ANNALS OF
MATHEMATICS
Hilbert’s fifth problem for local groups
By Isaac Goldbring
SECOND SERIES, VOL. 172, NO. 2
September, 2010
anmaah
Annals of Mathematics, 172 (2010), 1269–1314
Hilbert’s fifth problem for local groups
By I SAAC G OLDBRING
Abstract
We solve Hilbert’s fifth problem for local groups: every locally euclidean local
group is locally isomorphic to a Lie group. Jacoby claimed a proof of this in 1957,
but this proof is seriously flawed. We use methods from nonstandard analysis and
model our solution after a treatment of Hilbert’s fifth problem for global groups by
Hirschfeld.
1. Introduction
The most common version of Hilbert’s fifth problem asks whether every locally euclidean topological group is a Lie group. More precisely, if G is a locally
euclidean topological group, does there always exist a C ! structure on G such that
the group operations become C ! ? In 1952, Gleason, Montgomery, and Zippin
answered this question in the affirmative; see [3] and [11]. (“C ! ” means “real
analytic,” and by a C ! structure on a topological space X we mean a real analytic
manifold, everywhere of the same finite dimension, whose underlying topological
space is X.)
However, Lie groups, as first conceived by Sophus Lie, were not actually
groups but rather “local groups” where one could only multiply elements that were
sufficiently near the identity. (A precise definition of “local group” is given in the
next section.) It is thus natural to pose a local version of Hilbert’s fifth problem
(henceforth referred to as the Local H5): Is every locally euclidean local group
locally isomorphic to a Lie group? In [6], Jacoby claims to have solved the Local
H5 affirmatively. However, as pointed out by Plaut in [14], Jacoby fails to recognize
a subtlety in the way the associative law holds in local groups, as we will now
explain.
In our local groups, if the products xy; yz; x.yz/; and .xy/z are all defined,
then x.yz/ D .xy/z. This condition is called “local associativity” by Olver in [13].
A much stronger condition for a local group to satisfy is “global associativity” in
which, given any finite sequence of elements from the local group and two different
ways of introducing parentheses in the sequence, if both products thus formed exist,
then these two products are in fact equal. According to [13], “Jacoby’s Theorem 8,
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ISAAC GOLDBRING
which is stated without proof, essentially makes the false claim that local associativity implies global associativity.” Contrary to this “Theorem 8”, [13] has examples
of connected local Lie groups that are not globally associative. Since, according
to [13], “the rest of Jacoby’s paper relies heavily on his incorrect Theorem 8”, this
error invalidates his proof. Olver also indicates that one could probably rework
Jacoby’s paper to extract the truth of the Local H5. He suggests that this would be
a worthwhile endeavor, as, for example, the solution of Hilbert’s fifth problem for
cancellative semi-groups on manifolds heavily relies on the truth of the Local H5;
see [1] and [5].
Rather than reworking Jacoby’s paper, we have decided to mimic the nonstandard treatment of Hilbert’s fifth problem given by Hirschfeld [4]. Whereas
nonstandard methods simplified the solution of Hilbert’s fifth problem, such methods are even more natural in dealing with the Local H5. The nonstandard proof
of Hilbert’s fifth problem involves “infinitesimals”–elements in the nonstandard
extension of the group infinitely close to the identity; therefore, much of [4] goes
through in the local setting, since the infinitesimals form an actual group. These
infinitesimals are used to generate local 1-parameter subgroups, as in [16] and [4].
While our solution of the Local H5 is along the lines of the proof given by
Hirschfeld, many of the arguments have grown substantially in length due to the
care needed in working with local groups. We assume familiarity with elementary
nonstandard analysis; otherwise, consult [2] or [9]. We should also mention that
McGaffey [10] has a completely different nonstandard approach to Hilbert’s fifth
problem for local groups, using the connection between local Lie groups and finitedimensional real Lie algebras to reduce to a problem of approximating locally
euclidean local groups in a certain metric by local Lie groups. This approach has
not yet led to a solution.
I would like to thank Lou van den Dries for many helpful suggestions.
2. Preliminaries
In this section, we define the notion of local group and state precisely the
Local H5 (in two forms). We also introduce the fundamental objects in the entire
story, the local 1-parameter subgroups of G. After defining the local versions
of familiar constructions in group theory, e.g., morphisms between local groups,
sublocal groups, and local quotient groups, we prove some elementary properties
of local groups that are used heavily throughout the paper. We end this section by
fixing some notation concerning the nonstandard setting.
Throughout, m and n range over N WD f0; 1; 2; : : : g.
Statement of the local H5.
Definition 2.1. A local group is a tuple .G; 1; ; p/ where G is a Hausdorff
topological space with a distinguished element 1 2 G, and W ƒ ! G (the inversion
map) and p W  ! G (the product map) are continuous functions with open ƒ G
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HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
and open  G G, such that 1 2 ƒ, f1g G , G f1g , and for all
x; y; z 2 G:
(1) p.1; x/ D p.x; 1/ D x;
(2) if x 2 ƒ, then .x; .x// 2 , ..x/; x/ 2  and p.x; .x// D p..x/; x/ D 1;
(3) if .x; y/; .y; z/ 2  and .p.x; y/; z/; .x; p.y; z// 2 , then
(A)
p.p.x; y/; z/ D p.x; p.y; z//:
In the rest of this paper .G; 1; ; p/ is a local group; for simplicity we denote
it just by G. For the sake of readability, we write x 1 instead of .x/, and xy or
x y instead of p.x; y/. Note that .1/ D 1 and if x; y 2 ƒ, .x; y/ 2 , and xy D 1,
then y D x 1 and x D y 1 .
Definition 2.2.
(1) Let U be an open neighborhood of 1 in G. Then the restriction of G to U is
the local group GjU WD .U; 1; jƒU ; pjU /, where
ƒU WD ƒ \ U \ 1
.U / and U WD  \ .U U / \ p
1
.U /:
(2) G is locally euclidean if there is an open neighborhood of 1 homeomorphic
to an open subset of Rn for some n.
(3) G is a local Lie group if G admits a C ! structure such that the maps and p
are C ! .
(4) G is globalizable if there is a topological group H and an open neighborhood
U of 1H in H such that G D H jU .
By a restriction of G we mean a local group GjU where U is an open neighborhood of 1 in G.
Local H5-first form: If G is a locally euclidean local group, then some restriction of G is a local Lie group.
Local H5-second form: If G is a locally euclidean local group, then some
restriction of G is globalizable.
The equivalence of the two forms of the Local H5 is seen as follows. It is
known that if G is a local Lie group, then some restriction of G is equal to some
restriction of a Lie group. Thus the first form implies the second form. Conversely,
by the Montgomery-Zippin-Gleason solution to the original H5, if GjU is globalizable, the global group containing GjU will be locally euclidean and there will be
a C ! structure on it making it a Lie group. Then GjU will be a local Lie group.
Definition 2.3. A subgroup of G is a subset H G such that 1 2 H , H ƒ,
H H , and for all x; y 2 H , x 1 2 H and xy 2 H .
Note that then H is an actual group with product map pjH H . We will say
G has no small subgroups, abbreviated G is NSS, if there is a neighborhood U of
1 such that there are no subgroups H of G with H 6D f1g and H U: In analogy
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ISAAC GOLDBRING
with the solution to Hilbert’s fifth problem in the global case, we will first show
that every locally compact NSS local group has a restriction which is a local Lie
group and then we will prove that every locally euclidean local group is NSS.
The sets Un . As noted in the introduction, one obstacle in the local theory is
the potential lack of generalized associativity. The next definition captures the informal idea of being able to unambiguously multiply a finite sequence of elements.
Definition 2.4. Let a1 ; : : : ; an ; b 2 G with n 1. We define the notion
.a1 ; : : : ; an / represents b, denoted .a1 ; : : : ; an / ! b, by induction on n as follows:
.a1 / ! b if and only if a1 D b;
.a1 ; : : : ; anC1 / ! b if and only if for every i 2 f1; : : : ; ng, there exists bi0 ,
bi00 2 G such that .a1 ; : : : ; ai / ! bi0 , .ai C1 ; : : : ; anC1 / ! bi00 , .bi0 ; bi00 / 2 
and bi0 bi00 D b.
By convention, we say that .a1 : : : an / represents 1 when n D 0.
We will say a1 an is defined if there is b 2 G such that .a1 ; : : : ; an / represents b; in this case we will write a1 an for this (necessarily unique) b. If
a1 an is defined, then, informally speaking, all possible n-fold products are
defined and equal. If a1 an is defined and ai D a for all i 2 f1; : : : ; ng, we say
that an is defined and denote the unique product by an . In particular, a0 is always
defined and equals 1.
From now on, put An WD A
A
„ ƒ‚
…. Also, call W G symmetric if W ƒ
n times
and W D .W /. Note that U \ 1 .U / is a symmetric open neighborhood of 1 in
G for every open neighborhood U of 1 contained in ƒ .
We now prove a lemma which says that we can multiply any number of
elements unambiguously provided that the elements are sufficiently close to the
identity.
L EMMA 2.5. There are open symmetric neighborhoods Un of 1 for n > 0 such
that UnC1 Un and for all .a1 ; : : : ; an / 2 Un
n , a1 an is defined.
Proof. We let U1 be any open symmetric neighborhood of 1 and choose U2
to be a symmetric open neighborhood of 1 such that U2 U1 and U2 U2 .
Assume inductively that n 2 and that for m D 1; : : : ; n,
Um is a symmetric open neighborhood of 1;
UmC1 Um if m < n;
for all .a1 ; : : : ; am / 2 Um
m , a1 am is defined;
m ! G defined by ' .a ; : : : ; a / D a a is continuous.
the map 'm W Um
m 1
m
1
m
Let UnC1 be a symmetric open neighborhood of 1 such that UnC1 Un and
n
UnC1 'n 1 .U2 /. We show that this choice of UnC1 works. Let .a1 ; : : : ; anC1 / 2
.nC1/
UnC1 . We must show that .a1 ; : : : ; anC1 / represents a1 .a2 anC1 /. However, this is immediate by the inductive assumptions, using (A) several times. HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
1273
From now on, the sets Un will be as in the previous lemma. Also, if A Un ,
put An WD fa1 an j .a1 ; : : : ; an / 2 An g.
Local 1-parameter subgroups.
Definition 2.6. A local 1-parameter subgroup of G, henceforth abbreviated
local 1-ps of G, is a continuous map X W . r; r/ ! G, for some r 2 .0; 1, such
that
(1) image.X/ ƒ, and
(2) if r1 ; r2 ; r1 C r2 2 . r; r/, then .X.r1 /; X.r2 // 2  and
X.r1 C r2 / D X.r1 / X.r2 /:
Suppose X W . r; r/ ! G is a local 1-ps of G and s 2 . r; r/. It is easy to verify,
by induction on n, that if ns 2 . r; r/, then X.s/n is defined and X.ns/ D X.s/n .
We now define the local analog of the space which played a key role in the
solution of Hilbert’s fifth problem in the global setting.
Definition 2.7. Let X and Y be local 1-parameter subgroups of G. We say
that X is equivalent to Y if there is r 2 R>0 such that
r 2 domain.X / \ domain.Y /
and
X j. r; r/ D Y j. r; r/:
We let ŒX denote the equivalence class of X with respect to this equivalence relation. We also let L.G/ WD fŒX  j X is a local 1-ps of Gg.
In other words, L.G/ is the set of germs at 0 of local 1-parameter subgroups
of G. We often write X for an element of L.G/ and write X 2 X to indicate that
the local 1-ps X is a representative of the class X. We have a scalar multiplication
map .s; X/ 7! s X W R L.G/ ! L.G/, where s X is defined as follows. Let
X 2 X with X W . r; r/ ! G. If s D 0, then 0 X D O, where O W R ! G is
r
defined by O.t/ D 1 for all t 2 R and O D ŒO. Else, define sX W . jsjr ; jsj
/ ! G by
.sX /.t / D X.st/. Then sX is a local 1-ps of G, and we set s X D ŒsX . It is easy
to verify that for all X 2 L.G/ and s; s 0 2 R, 1 X D X and s .s 0 X/ D .ss 0 / X.
Suppose that X1 and X2 are local 1-parameter subgroups of G with ŒX1  D
ŒX2 . Suppose also that t 2 domain.X1 / \ domain.X2 /. Then X1 .t / D X2 .t /. To
see this, let r 2 R>0 be such that X1 j. r; r/ D X2 j. r; r/ and choose n > 0 such
that nt 2 . r; r/. Then X1 .t / D .X1 . nt //n D .X2 . nt //n D X2 .t /. Hence, it makes
sense to define, for X 2 L.G/,
[
domain.X/ WD
domain.X /
X 2X
and for t 2 domain.X/, we will write X.t / to denote X.t / for any X 2 X with
t 2 domain.X/.
The category of local groups. In this subsection, we define the analogs of a
few ordinary group theoretic notions in the local group setting. We will not need
most of this material until Section 8. The presentation given here borrows from [12]
and [15].
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ISAAC GOLDBRING
Definition 2.8. Suppose G D .G; 1; ; p/ and G 0 D .G 0 ; 10 ; 0 ; p 0 / are local
groups with domain./ D ƒ, domain.p/ D , domain.0 / D ƒ0 , and domain.p 0 / D
0 . A morphism from G to G 0 is a continuous function f W G ! G 0 such that:
(1) f .1/ D 10 , f .ƒ/ ƒ0 , and .f f /./ 0 ,
(2) f ..x// D 0 .f .x// for x 2 ƒ, and
(3) f .p.x; y// D p 0 .f .x/; f .y// for .x; y/ 2 .
Suppose U1 and U2 are open neighborhoods of 1 in G and suppose that f1 W
GjU1 ! G 0 and f2 W GjU2 ! G 0 are both morphisms. We say that f1 and f2 are
equivalent if there is an open neighborhood U3 of 1 in G such that U3 U1 \ U2
and f1 jU3 D f2 jU3 . This defines an equivalence relation on the set of morphisms
f W GjU ! G 0 , where U ranges over all open neighborhoods of 1 in G. We will
call the equivalence classes local morphisms from G to G 0 . To indicate that f is a
local morphism from G to G 0 , we write “f W G ! G 0 is a local morphism,” even
though f is not a function from G to G 0 , but rather an equivalence class of certain
partial functions.
Suppose that G, G 0 , and G 00 are local groups and f W G ! G 0 and g W G 0 !
G 00 are local morphisms. Choose a representative of f whose image lies in the
domain of a representative of g. Then these representatives can be composed and
the composition is a morphism. We denote by g ı f the equivalence class of
this composition. The equivalence class of the identity function on G is a local
morphism and will be denoted by idG . We thus have the category LocGrp whose
objects are the local groups and whose arrows are the local morphisms with the
composition defined above. A local isomorphism is an isomorphism in the category
LocGrp. Hence, if f W G ! G 0 is a local isomorphism, then there is a representative
f W GjU ! G 0 jU 0 , where U and U 0 are open neighborhoods of 1 and 10 in G and
G 0 respectively, f W U ! U 0 is a homeomorphism, and f W GjU ! G 0 jU 0 and
f 1 W G 0 jU 0 ! GjU are morphisms. If there exists a local isomorphism between
the local groups G and G 0 , then we say that G and G 0 are locally isomorphic.
Example 2.9. One can consider the additive group R as a local group and
. r; r/ for r 2 .0; 1 as a restriction of this local group, so that a local 1-ps
. r; r/ ! G is a morphism of local groups. In this way, the elements of L.G/ are
local morphisms from R to G. Moreover, given any local morphism f W G ! G 0 ,
one gets a map of sets L.f / W L.G/ ! L.G 0 / given by L.f /.X/ D f ı X. It can
easily be checked that this yields a functor L W LocGrp ! Sets.
Definition 2.10. A sublocal group of G is a set H G containing 1 for which
there exists an open neighborhood V of 1 in G such that
(1) H V and H is closed in V ;
(2) if x 2 H \ ƒ and x
1
2 V , then x
1
2 H;
(3) if .x; y/ 2 .H H / \  and xy 2 V , then xy 2 H .
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HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
With H and V as above, we call H a sublocal group of G with associated neighborhood V . A normal sublocal group of G is a sublocal group H of G with an
associated neighborhood V such that V is symmetric and
(4) if y 2 V and x 2 H are such that yxy
yxy 1 2 H .
1
is defined and yxy
1
2 V , then
For such H and V , we also say that H is a normal sublocal group of G with
associated normalizing neighborhood V .
One checks easily that if H is a sublocal group of G with associated neighborhood V , then
.H; 1; jH \ ƒ \ 1
.V /; pj.H H / \  \ p
1
.V //
is a local group, to be denoted by H for simplicity, and that then the inclusion
H ,! G is a morphism. Sublocal groups H and H 0 of G are equivalent if there is
an open neighborhood U of 1 in G such that H \ U D H 0 \ U .
Example 2.11. Suppose that f W G ! G 0 is a morphism and ƒ D G. Then
ker.f / WD f 1 .f1g/ is a normal sublocal group of G with associated normalizing
neighborhood G.
The following lemma has a routine verification.
L EMMA 2.12. Suppose H is a normal sublocal group of G with associated
normalizing neighborhood V . Suppose U H is open in H and symmetric. Let
U 0 V be a symmetric open neighborhood of 1 in G such that U D H \ U 0 .
Then H jU is a normal sublocal group of G with associated normalizing neighborhood U 0 .
We now investigate quotients in this category. For the rest of this subsection, assume that H is a normal sublocal group of G with associated normalizing
neighborhood V .
L EMMA 2.13. Let W be a symmetric open neighborhood of 1 in G such that
W U6 and W 6 V . Then:
(1) The binary relation EH on W defined by
EH .x; y/ if and only if x
1
y2H
is an equivalence relation on W .
(2) For x 2 W , let xH WD fxh j h 2 H and .x; h/ 2 g. Then for x; y 2 W ,
EH .x; y/ holds if and only if .xH / \ W D .yH / \ W . In other words, if
EH .x/ denotes the equivalence class of x, then EH .x/ D .xH / \ W . We call
the equivalence classes local cosets of H .
Proof. For (1), first note that the reflexivity of EH is trivial. Now suppose
EH .x; y/ holds, i.e., x 1 y 2 H . Since .x 1 y/ 1 2 V , we have that .x 1 y/ 1 2 H .
It can easily be verified that .x 1 y/ 1 D y 1 x, so EH .y; x/ holds. Finally,
1276
ISAAC GOLDBRING
suppose EH .x; y/ and EH .y; z/ hold. It follows from .x 1 y/.y 1 z/ 2 V that
.x 1 y/.y 1 z/ 2 H . However, since x; y; z 2 U4 , we know that .x 1 y/.y 1 z/ D
x 1 z, so we have transitivity and (1) is proven.
For (2), first suppose that EH .x; y/ holds. Let w 2 .yH / \ W . Let h 2 H
be such that w D yh. Then h D y 1 w 2 W 2 . Since .x 1 y/h 2 V , we have
that .x 1 y/h 2 H . But .x 1 y/h D x 1 w, so w 2 .xH / \ W . By symmetry,
.xH / \ W D .yH / \ W . For the converse, suppose that .xH / \ W D .yH / \ W .
Since y 2 .yH / \ W , we have that y 2 .xH / \ W . Let h 2 H be such that y D xh.
Then since xh 2 W , we know that x 1 .xh/ is defined. We thus have, by (A), that
x 1 y D h 2 H and so EH .x; y/ holds.
Let H;W W W ! W =EH be the canonical map. Give W =EH the quotient
topology. Then H;W becomes an open continuous map.
Let H;W W W=EH ! W =EH be defined by H;W .EH .x// D EH .x 1 /. Let
H;W WD .H;W H;W /..W W / \ p 1 .W //. It is easy to check that one
can define a map pH;W W H;W ! W =EH by pH;W .EH .x/; EH .y// D EH .xy/,
where x and y are representatives of their respective local cosets chosen so that
xy 2 W . The following lemma is easy to verify.
L EMMA 2.14. With the notation as above,
.G=H /W WD .W =EH ; EH .1/; H;W ; pH;W /
is a local group and H;W W GjW ! .G=H /W is a morphism.
In the construction of quotients, we have made some choices. We would
expect that, locally, we have the same local group. The following lemma expresses
this.
L EMMA 2.15. Suppose that H 0 is also a normal sublocal group of G with
associated normalizing neighborhood V 0 such that H is equivalent to H 0 . Let W 0
be a symmetric open neighborhood of 1 in G used to construct .G=H 0 /W 0 . Then
.G=H /W and .G=H 0 /W 0 are locally isomorphic.
Proof. Let U be an open neighborhood of 1 in G such that H \ U D H 0 \ U .
Choose open U1 containing 1 such that U12 U . Let U2 WD W \W 0 \U1 . Consider
the map W H;W .U2 / ! W 0 =EH 0 given by .EH .x// D EH 0 .x/ for x 2 U2 .
One can verify that W .G=H /W jH;W .U2 / ! .G=H 0 /W 0 induces the desired
local isomorphism.
From now on, we write G=H to denote .G=H /W , where W is any open
neighborhood of 1 as in Lemma 2.13. We will also write W G ! G=H to denote
the local morphism induced by H;W W GjW ! .G=H /W . Since all of the various
quotients defined in this way are locally isomorphic, fixing one’s attention on one
particular local coset space is no loss of generality for our purposes.
Further properties of local groups. In this subsection, we first show that we
can place two further assumptions on our local groups which lead to no loss of
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HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
generality in the solution of the Local H5. Under these new assumptions, we
derive several properties of the notion “a1 an is defined.”
Let U WD ƒ \ 1 .ƒ/, an open neighborhood of 1 in G. Note that if g 2 U ,
then .g/ 2 ƒ and ..g// D g 2 ƒ, implying that ƒU D U .
From this point on, we assume that ƒ D G for every local group G in this
paper. By the preceding remarks, every local group has a restriction satisfying this
condition and so this is no loss of generality for our purpose. Note that, with these
assumptions, if .x; y/ 2  and xy D 1, then x D y 1 and y D x 1 . In particular,
.x 1 / 1 D 1 for all x 2 G, and G is symmetric.
The following lemma is obvious for topological groups, but requires some
care in the local group setting.
L EMMA 2.16 (Homogeneity).
(1) For any g 2 G, there are open neighborhoods V and W of 1 and g respectively
such that fgg V , gV W , fg 1 g W , g 1 W V , and the maps
v 7! gv W V ! W and w 7! g 1 w W W ! V are each others inverses (and
hence homeomorphisms).
(2) G is locally compact if and only if there is a compact neighborhood of 1.
Proof. Clearly (1) implies (2). For any g 2 G, define
g WD fh 2 G j .g; h/ 2 g:
Then g is an open subset of G and Lg W g ! G, defined by Lg .h/ D gh, is
continuous. Let V WD .Lg / 1 .g 1 /. Then V is open and 1 2 V . Let W D
Lg .V / g 1 : Then W is open since W D Lg 1 1 .V /. From our choices we can
check that Lg jV and Lg 1 jW are inverses of each other.
C OROLLARY 2.17. Let U D U3 . Then for any g; h 2 U such that .g; h/ 2 U ,
one has .h 1 ; g 1 / 2 U and .gh/ 1 D h 1 g 1 .
Proof. Suppose g; h 2 U with .g; h/ 2 U . Then we have h
Hence h 1 g 1 .gh/ is defined and
h
1
g
1
.gh/ D h
1
.g
1
1
.gh// D h
by (A). But h 1 g 1 .gh/ D .h 1 g 1 / .gh/, so h
h 1 g 1 D .gh/ 1 2 U , we have .h 1 ; g 1 / 2 U .
1g 1
1 ; g 1 ; gh
2 U.
hD1
D .gh/
1.
Also, since
From now on, we make the following further assumption on our local group G:
if .g; h/ 2 , then .h
1
;g
1
/ 2  and .gh/
1
Dh
1
g
1
:
By the preceding corollary, every local group has a restriction satisfying this assumption and so this is no loss of generality for our purpose.
L EMMA 2.18. Let a; a1 ; : : : ; an 2 G. Then:
(1) If a1 an is defined and 1 i j n, then ai aj is defined. In particular,
if an is defined and m n, then am is defined.
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ISAAC GOLDBRING
(2) If am is defined and i; j 2 f0; : : : ; mg are such that i Cj D m, then .ai ; aj / 2 
and ai aj D am .
(3) If an is defined and, for all i; j 2 f1; : : : ; ng with i C j D n C 1, one has
.ai ; aj / 2 , then anC1 is defined. More generally, if a1 an is defined,
ai anC1 is defined for all i 2 f2; : : : ; ng and .a1 ai ; ai C1 anC1 / 2 
for all i 2 f1; : : : ; ng, then a1 anC1 is defined.
(4) If an is defined, then .a 1 /n is defined and .a 1 /n D .an / 1 . (In this case, we
denote .a 1 /n by a n .) More generally, if a1 an is defined, then an 1 a1 1
is defined and .a1 an / 1 D an 1 a1 1 .
(5) If k; l 2 Z, l 6D 0, and akl is defined, then ak is defined, .ak /l is defined, and
.ak /l D akl .
Proof. The proofs of (1) and (2) are immediate from the definitions. (3) follows from repeated uses of (A). We prove the first assertion of (4) by induction on n.
The cases n D 1 and n D 2 are immediate from our assumptions on local groups. For
the induction step, suppose anC1 is defined and i; j 2 f1; : : : ; ng with i Cj D nC1.
Then since .aj ; ai / 2 , by induction we have that ..a 1 /i ; .a 1 /j / 2 . Hence,
by (3) of the lemma, we know that .a 1 /nC1 is defined. Also,
.a
1 nC1
/
D .a
1 n
/ a
1
D .an /
1
a
1
D .a an /
1
D .anC1 /
1
:
The second assertion of (4) is proved in the same manner.
We now prove (5). It is easy to see that it is enough to check the assertion for
k; l 2 N, l 6D 0. Since k k l, we have that ak is defined by (1). We prove the
rest by induction on l. This is clear for l D 1. Suppose the assertion is true for
all i l and suppose that ak.lC1/ is defined. To see that .ak /lC1 is defined, we
must check that ..ak /i ; .ak /j / 2  for i; j 2 f1; : : : ; lg such that i C j D l C 1.
This is the case by (2), since for such i; j , we have k i C k j D k .l C 1/ and
by induction .ak /i D aki and .ak /j D akj . Now that we know that .ak /lC1 is
defined, we must have, by induction, that
.ak /lC1 D .ak /l ak D akl ak D aklCk :
By (4) of the previous lemma, we can now speak of ak being defined for any
k 2 Z, where ak being defined for k < 0 means that a k is defined.
C OROLLARY 2.19. Suppose i; j 2 Z and i j < 0. If ai and aj are defined
and .ai ; aj / 2 , then ai Cj is defined and ai aj D ai Cj .
Proof. The fact that aiCj is defined is clear from the preceding lemma. Next
note that the result is trivial if i D j . We only prove the case that i > 0 and
j < 0 and i > jj j, the other cases being similar. By the preceding lemma, we have
ai D ai Cj a j , whence ai aj D ai Cj by (A).
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
1279
The nonstandard setting. We assume familiarity with this setting; see [2] and
[9] for details. Here we just fix notations and terminology. To each relevant “basic”
set S corresponds functorially a set S S , the nonstandard extension of S . In
particular, N; R; G extend to N ; R ; G , respectively. Also, any (relevant) relation
R and function F on these basic sets extends functorially to a relation R and
function F on the corresponding nonstandard extensions of these basic sets. For
example, the linear ordering < on N extends to a linear ordering < on N , and the
local group operation p W  ! G of G extends to an operation p W  ! G . For
the sake of readability we only use a star in denoting the nonstandard extension
of a basic set, but drop the star when indicating the nonstandard extension of a
relation or function on these basic sets. For example, when x; y 2 R we write
x C y and x < y rather than x C y and x < y.
Given an ambient Hausdorff space S and s 2 S , the monad of s, notation:
.s/, is by definition the intersection of all U S with U a neighborhood of
s in S ; the elements of .s/ are the points of the nonstandard space S that are
infinitely close to s. The points of S that are infinitely close to some s 2 S are
is the set of nearstandard points of S :
called nearstandard, and Sns
[
Sns
D
.s/:
s2S
.s/ \ .s 0 /
,
Since S is Hausdorff,
D ∅ for distinct s; s 0 2 S . Thus, for s 2 Sns
we can define the standard part of s , notation: st.s /, to be the unique s 2 S such
, we put s s if and only if st.s / D st.s /.
that s 2 .s/. For s1 ; s2 2 Sns
1
2
1
2
We let WD .1/ be the monad of 1 in G ; note that is a group with product
map .x; y/ 7! xy WD p.x; y/. From now on, we let ; ; ; , and N range over N
and i and j range over Z , but m and n will always denote elements of N. We use
the Landau notation as follows: i D o./ means ji j < n for all n > 0 and i D O./
means jij < n for some n > 0.
In our nonstandard arguments, we will assume as much saturation as needed.
The following lemma uses routine nonstandard methods and is left for the reader
to verify.
L EMMA 2.20. G is NSS if and only if there are no internal subgroups of other than f1g.
We also consider the internal versions of the notions and results that appear
earlier in this section. In particular, if .a1 ; : : : ; a / is an internal sequence in G ,
then it makes sense to say that .a1 ; : : : ; a / (internally) represents b, where b 2 G .
We say that a1 a is defined if there exists b 2 G such that .a1 ; : : : ; a / represents b. Likewise, we have the notion “ai is defined” for a 2 G ; i 2 Z .
The proof of the following lemma is trivial.
L EMMA 2.21.
(1) Suppose a; b 2 G, a0 2 .a/, and b 0 2 .b/. If .a; b/ 2 , then .a0 ; b 0 / 2  ,
, and st.a0 b 0 / D a b.
a0 b 0 2 Gns
1280
ISAAC GOLDBRING
and b 2 , .a; b/; .b; a/ 2  , a b; b a 2 G , and st.a b/ D
(2) For any a 2 Gns
ns
st.b a/ D st.a/.
, if .a; b
(3) For any a; b 2 Gns
a0
1/
2  and ab
(4) For any a 2 G, 2 .a/, and any n, if
and .a0 /n 2 .an /.
an
1
2 , then a b.
is defined, then .a0 /n is defined
A standard consequence of the last item of the preceding lemma is that the
partial function pn W G * G, defined by pn .a/ D an if an is defined, has an open
domain and is continuous. Note that Un domain.pn /.
L EMMA 2.22. Suppose U is a neighborhood of 1 in G and a 2 . Then there
is a > N such that a is defined and a 2 U for all 2 f1; : : : ; g.
Proof. Let X D f 2 N j a is defined and a 2 U g. Then X is an internal
subset of N which contains N since is a subgroup of G . Hence, by overflow,
there is a > N such that f0; 1; : : : ; g X .
3. Connectedness and purity
In this section, we describe the different ways powers of infinitesimals can
grow and its impact on the existence of connected subgroups of G. We also show
how to build local 1-parameter subgroups from “pure” infinitesimals.
In the rest of this paper, we assume that G is locally compact. In particular,
G is regular as a topological space. Also, a and b will denote elements of G .
For > N, we set:
G./ WD fa 2 j ai is defined and ai 2 for all i D o./g;
G o ./ WD fa 2 j ai is defined and ai 2 for all i D O./g:
Note that G o ./ G./ and both sets are symmetric.
L EMMA 3.1. Let a 2 and > N. Then the following are equivalent:
(1) ai is defined and ai 2 for all i 2 f1; : : : ; g;
(2) a 2 G o ./;
(3) there is 2 f1; : : : ; g such that D O. / and ai is defined and ai 2 for all
i 2 f1; : : : ; g.
Proof. (1) ) (2): Fix such that D O./. We claim that if a is defined,
then a 2 . To see this, write D n C for some n and < . By Lemma 2.18,
a D .a /n a 2 :
We now prove by internal induction that ai is defined for all i 2 f1; : : : ; g. The
assertion is true for i D 1 and inductively suppose that ai is defined and i C 1 .
By Lemma 2.18, in order to show that ai C1 is defined, it suffices to show that
.ak ; al / 2  for all k; l 2 f1; : : : ; i g with k C l D i C 1. However, .ak ; al / 2
 , finishing the induction.
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
1281
Clearly (2)) (1) and (3), so it remains to prove that (3) ) .2/. So assume that
there is 2 f1; : : : ; g as in (3). By the proof of (1)) .2/, we see that a 2 G o . /.
Since G o ./ G o ./, we are finished.
Definition 3.2. We say that a 2 is degenerate if, for all i , ai is defined and
ai 2 .
It is easy to see that G is NSS if and only if has no nondegenerate elements
other than 1.
L EMMA 3.3. Let a1 ; : : : ; a be an internal sequence of elements of G with
> 0 such that for all i 2 f1; : : : ; g we have that ai 2 , a1 ai is defined, and
. Then the set
a1 ai 2 Gns
S WD fst.a1 ai / W 1 i g G
is compact and connected (and contains 1).
The proof is as in the global case, but we include it for completeness.
Proof. Suppose that for every compact K G, there is i 2 f1; : : : ; g such that
a1 ai … K . Then, by saturation, there is i 2 f1; : : : ; g such that a1 ai … K for all compact K G, contradicting local compactness. Hence there is compact
K G with a1 ai 2 K for all i 2 f1; : : : ; g, so S K. It is well-known that
S must be closed, being the standard part of an internal nearstandard set. Hence S
is compact.
Suppose S is not connected. Then we have disjoint open subsets U and V
of G such that S U [ V , S \ U 6D ∅, and S \ V 6D ∅. Assume 1 2 U , so
a1 2 U . Choose i 2 f1; : : : ; g minimal such that a1 ai 2 V . Then i 2
and a1 ai 1 2 U . Now a WD st.a1 ai 1 / D st.a1 ai / 2 S . If a 2 U ,
then a1 ai 1 2 U and if a 2 V , then a1 ai 2 V ; either option yields a
contradiction.
Until further notice, U denotes a compact symmetric neighborhood of 1 such
that U U2 .
Definition 3.4. Let a 2 G . If, for all i , ai is defined and ai 2 U , define
ordU .a/ D 1. Else, define ordU .a/ D if, for all i with ji j , ai is defined,
ai 2 U , and is the largest element of N for which this happens.
It is easy to see that ordU .a/ D 0 if and only if a … U . By Lemma 2.22, we
can see that ordU .a/ > N if a 2 . Note that if a 2 and ordU .a/ D 2 N ,
then aC1 is defined. To see this, it is enough to check that .ai ; aj / 2  for all
i; j 2 f1; : : : ; g such that i C j D C 1. However, this is true since
.ai ; aj / 2 U U U2 U2  :
Hence, if ordU .a/ D , then this is not because aC1 is undefined but rather for the
more important reason that aC1 is defined but not in U . We have thus captured
the correct adaptation of the global notion to our local setting.
1282
ISAAC GOLDBRING
L EMMA 3.5. Suppose a 2 and ai … for some i D o.ordU .a//. Then U
contains a nontrivial connected subgroup of G.
Proof. By the previous lemma, the set
GU .a/ WD fst.ai / j ji j D o.ordU .a//g
is a union of connected subsets of U , each containing 1, hence is connected. It is
also a subgroup of G by Lemma 2.18 and Corollary 2.19.
Definition 3.6. We say that a 2 is U -pure1 if it is nondegenerate and a 2
G. /, where WD ordU .a/. We say that a 2 is pure if it is V -pure for some
compact symmetric neighborhood V of 1 such that V U2 .
The last lemma now reads that if U contains no nontrivial connected subgroup
of G, then every a 2 which is nondegenerate is U -pure.
L EMMA 3.7. Let a 2 . Then a is pure if and only if there is > N such that
a is defined, a … , and a 2 G./.
Proof. First suppose that a is V -pure. Let D ordV .a/. Then certainly a is
defined. Now since aC1 is defined, a … , else aC1 D a a 2 , contradicting
aC1 … V . However, ai 2 for i D o./ by definition of V -purity. Conversely,
suppose one has > N such that a is defined, a … , and a 2 G./. Since a … ,
there is a compact symmetric neighborhood V of 1 such that V U2 and such
that a … V . Let WD ordV .a/. Then < , implying that a 2 G. / and thus a
is V -pure.
Let Q range over internal subsets of G such that Q , 1 2 Q, and Q
is symmetric. If is such that for every internal sequence a1 ; : : : ; a from Q,
a1 a is defined, then we define Q to be the internal subset of G consisting
of all a1 a for a1 ; : : : ; a an internal sequence from Q. In this situation, we
say that Q is defined.
Definition 3.8. If for all , Q is defined and Q , then we say that Q is
degenerate. If Q is defined and Q U for all , define ordU .Q/ D 1. Else,
define ordU .Q/ D if Q is defined, Q U , and is the largest element of
N for which this happens.
Using an overflow argument similar to that in Lemma 2.22, we have that
ordU .Q/ > N. One can also check that if ordU .Q/ 2 N , then QordU .Q/C1 is
defined.
L EMMA 3.9. Suppose Q ª for some D o.ordU .Q//. Then U contains
a nontrivial connected subgroup of G.
1 We
have decided to replace the use of “regular" in [4] with “pure" as “regular" already has a
different meaning for topological spaces.
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
1283
Proof. As in the proof of Lemma 3.5, the set
GU .Q/ D fst.a/ j a 2 Q for some D o.ordU .Q//g
is a union of connected subsets of U , each containing 1, and thus itself is a connected subset of U . To see that GU .Q/ is a group, suppose that a 2 Q and
b 2 Q for ; D o.ordU .Q//. Then C D o.ordU .Q// and so the concatenation
of the sequences representing a and b represents an element of G , which must
necessarily be a b. Hence a b 2 QC and st.a/ st.b/ D st.a b/ 2 GU .Q/. That
GU .Q/ is closed under inverses is an easy consequence of Lemma 2.18.
Definition 3.10. We say that Q is U -pure if Q is nondegenerate and if Q for all D o.ordU .Q//. We say that Q is pure if it is V -pure for some compact
symmetric neighborhood V of 1 in G such that V U2 .
The previous lemma now reads that if U contains no nontrivial connected
subgroups of G, then every nondegenerate Q as above is U -pure.
L EMMA 3.11. Q is pure if and only if there is some > N such that Q is
defined, Q ª , and Q for all D o. /.
Proof. First suppose Q is V -pure. Let D ordV .Q/. Then > N, Q is
defined, Q ª (otherwise Q C1 V , contradicting the definition of ),
and Q for D o. / by definition of being V -pure. Conversely, suppose we
have > N such that Q is defined, Q ª , and Q for D o. /. Since
Q ª , we can find a compact symmetric neighborhood V of 1 in G such that
V U2 and Q ª V . Let WD ordV .Q/. Then < , implying that Q for
all D o./, that is Q is V -pure.
Definition 3.12. G has no small connected subgroups, henceforth abbreviated
G is NSCS, if there is a neighborhood of 1 in G that contains no connected subgroup of G other than f1g. We say that G is pure if every nondegenerate Q as
above is pure.
C OROLLARY 3.13. If G is NSCS, then G is pure.
Pure infinitesimals and local 1-parameter subgroups. Suppose that a 2 is
U -pure and 0 < D O. /, where WD ordU .a/. Let
†;a;U WD fr 2 R>0 j aŒr is defined and ai 2 U if ji j Œrg:
Since D O./, †;a;U 6D ∅. Let r;a;U D sup †;a;U , where we let r;a;U D 1
if †;a;U D R>0 . Suppose s < r;a;U . Then s 2 †;a;U and aŒs 2 U . Since
aŒsC1 is defined (see comment after Definition 3.4; all that was used there was
. Hence we have that aŒs
that U U ), we have that aŒsC1 D aŒs a 2 Gns
is defined and nearstandard for s 2 . r;a;U ; 0/ as well since, for such s, we have
that Œs D Œ. s/ or Œ. s/ 1.
L EMMA 3.14. The map X W . r;a;U ; r;a;U / ! G given by X.s/ WD st.aŒs /
is a local 1-ps of G.
1284
ISAAC GOLDBRING
Proof. Throughout this proof, we let r WD r;a;U . We first show that X
satisfies the additivity condition in the definition of a local 1-ps of G. Note that the
preceding arguments show that image.X / U2 . Suppose first that s1 ; s2 ; s1 C s2 2
.0; r/. Since Œs1 CŒs2  D Œ.s1 Cs2 / or Œ.s1 Cs2 / 1, we have that aŒs1 CŒs2 
is defined and st.aŒ.s1 Cs2 / / D st.aŒs1 CŒs2  /. Hence X.s1 C s2 / D X.s1 / X.s2 /.
The case that s1 ; s2 ; s1 Cs2 2 . r; r/ with s1 s2 < 0 works similarly, using Corollary
2.19. Using the fact that X. s/ D X.s/ 1 , we get the additivity property for
s1 ; s2 2 . r; 0/ for which s1 C s2 2 . r; 0/.
It remains to show that X is continuous. By Lemma 2.16, it is enough to show
that X is continuous at 0. Let V be a compact neighborhood of 1 in G. If i D o./,
then ai is defined and ai 2 V . By saturation, we have n such that n1 < r and
such that if j i j < n1 , then ai is defined and ai 2 V . Hence X.s/ 2 V for jsj < n1 . We end this section with an extension lemma which will be used later in the
paper. From now on, I WD Œ 1; 1 R.
L EMMA 3.15. Suppose X W I ! G is a continuous function such that for all
r; s 2 I , if r C s 2 I , then .X.r/; X.s// 2  and X.r C s/ D X.r/ X.s/. Further
suppose that X.I / U4 . Then there exists " 2 R>0 and a local 1-parameter
subgroup Xx W . 1 "; 1 C "/ ! G of G such that Xx jI D X.
x D X and, for
Proof. Fix " 2 .0; 12 /. Define Xx W . 1 "; 1 C "/ ! G by XjI
x
x
x
0 < ı < ", X.1 C ı/ D X.1/ X.ı/ and X . 1 ı/ D X .1 C ı/ 1 . We claim that
such an Xx satisfies the conclusion of the lemma. By Lemma 2.16, it is enough to
x
x
show that .X.r/;
X.s//
2  and Xx .r/Xx .s/ D Xx .r C s/ for all r; s 2 . 1 "; 1 C "/
for which r C s 2 . 1 "; 1 C "/. We first note that Xx .r/; Xx .s/ 2 U24 so that
x
.Xx .r/; X.s//
2  is immediate. The additivity property of Xx splits up into several
cases.
Case 1a: r; s; r C s 2 Œ0; 1. This case is trivial.
Case 1b: r; s 2 Œ0; 1 C "/, r C s 2 .1; 1 C "/. Then
x C s/ D X.1/X.r C s
X.r
1/ D X.1/X.r
1/X.s/ D Xx .r/Xx .s/:
Case 2a: r; s; r C s 2 Œ 1; 0. This case is trivial.
Case 2b: r; s 2 . 1
"; 0, r C s 2 . 1
x C s/ D Xx . s
X.r
r/
1
"; 1/. Then
D .Xx . s/Xx . r//
1
D Xx .r/Xx .s/:
Case 3a: r 2 Œ0; 1, s 2 Œ 1; 0. This case is trivial.
Case 3b: r 2 Œ0; 1, s 2 . 1
"; 1/. Then
Xx .r C s/ D Xx . r
s/
D .X.1/X. r
1
s
1//
D X.r C s C 1/X. 1/
D X.r/X.s C 1/X. 1/
1
1285
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
D X.r/.X.1/X. s
D Xx .r/Xx .s/:
1
1//
Case 3c: r 2 .1; 1 C "/, s 2 Œ 1; 0. Then by Case 3b, we have
x C s/ D Xx . s
X.r
r/
1
D .Xx . s/Xx . r//
1
D Xx .r/Xx .s/:
Case 3d: r 2 .1; 1 C "/, s 2 . 1 "; 1/. For this case, we will need the
observation that for any ı 2 . "; "/, we have X.1/X.ı/ D X.ı/X.1/. We shall
only prove this for ı 0, the proof being similar for ı < 0. The following chain
of equalities verifies the observation:
X.1/X.ı/ D X.1
D X.1
ı/X.ı/X.ı/
ı/X.2ı/
D X.ı/X.1
Now, we have
2ı/X.2ı/
D X.ı/X.1/:
x C s/ D X.r C s/X.1/X. 1/
X.r
D X.1/X.r C s/X. 1/
D X.1/X.r 1/X.s C 1/X. 1/
D Xx .r/Xx .s/:
Case 4: r 2 . 1 "; 0, s 2 Œ0; 1C"/. This case is treated exactly like Case 3. 4. Consequences of NSS
In this section we assume G is NSS.
Special neighborhoods. The next lemma is from [8], and we repeat its proof,
with a more detailed justification of some steps.
L EMMA 4.1. There is a neighborhood V of 1 such that V U2 and for all
x; y 2 V , if x 2 D y 2 , then x D y.
Proof. Choose a compact symmetric neighborhood U of 1 in G with U U6
such that U contains no subgroups of G other than f1g. Choose a symmetric open
neighborhood W of 1 in G such that W 5 U . Since U is compact, we can find
a symmetric open neighborhood V of 1 in G such that V 2 W and gVg 1 W
for all g 2 U . We claim that this V fulfills the desired condition.
Suppose that x; y 2 V are such that x 2 D y 2 . Let a WD x 1 y 2 V 2 W U .
Since V U5 , we have axa D .x 1 y/x.x 1 y/ D x 1 y 2 D x 1 x 2 D x.
C LAIM. For all n, an is defined, an 2 U , and an D xa
nx 1.
We prove the claim by induction. The case n D 1 follows from the fact that
axa D x. For the inductive step, suppose the assertion holds for all m 2 f1; : : : ; ng.
In order to show that anC1 is defined, it suffices to show that .ai ; aj / 2  for all
1286
ISAAC GOLDBRING
i; j 2 f1; : : : ; ng such that i C j D n C 1. However, by the inductive hypothesis,
.ai ; aj / 2 U U , proving that anC1 is defined. Since U U6 , we have
anC1 D an a D .xa
n
x
1
/.xa
1
x
1
/ D xa
n 1
x
1
:
It remains to show that anC1 2 U . First suppose that n C 1 is even, say n C 1 D 2m.
Then
anC1 D am am D xa m x 1 am 2 xW W 2 U:
Now if n C 1 is odd, then anC1 D an a 2 W 2 W 2 U .
This finishes the proof of the claim. The claim yields a subgroup aZ of G
with aZ U . Thus a D 1 and so x D y.
Definition 4.2. A special neighborhood of G is a compact symmetric neighborhood U of 1 in G such that U U2 , U contains no nontrivial subgroup of G,
and for all x; y 2 U, if x 2 D y 2 , then x D y.
By the last lemma, we know that G has a special neighborhood. In the rest
of this section, we fix a special neighborhood U of G. Then every a 2 n f1g is
U-pure. For a 2 G , we set ord.a/ WD ordU .a/:
L EMMA 4.3. Let a 2 G . Then ord.a/ is infinite if and only if a 2 .
Proof. We remarked earlier that if a 2 , then ord.a/ is infinite. Now suppose
ord.a/ is infinite. Then, for all k 2 Z, ak is defined and ak 2 U . It is easy to
prove by induction that st.a/k is defined and st.a/k 2 U for all k 2 Z. This implies
that st.a/ D 1 since U is a special neighborhood.
L EMMA 4.4. Suppose > N and a 2 G. /. Then:
(1) if a 6D 1, then a is U-pure and D O.ord.a//;
(2) if i D o./, then ai is defined and ai 2 ;
(3) Let †a WD †;a;U and ra WD r;a;U . Let Xa W . ra ; ra / ! G be defined by
X.s/ WD st.aŒs  /. Then Xa is a local 1-ps of G.
Proof. It has already been remarked why a is U-pure. If ord.a/ D o. /, then
aord.a/ 2 , a contradiction. Hence, D O.ord.a//. By (1), if i D o. /, then
i D o.ord.a//, whence ai is defined and ai 2 since a is U-pure. This proves (2).
(3) follows immediately from Lemma 3.14.
C OROLLARY 4.5. Suppose G is not discrete. Then L.G/ 6D fOg.
Proof. Take a 2 n f1g. Set WD ord.a/. Since a 2 , we must have > N.
Since a 2 G./, we have ŒXa  2 L.G/, where Xa is as defined in Lemma 4.4.
We need to show that ŒXa  6D O. It suffices to show that for every n such that
1
1
Œn

… . Let t WD Œ n1 . Then t D n1 ", with " 2 Œ0; 1/ . Towards
n 2 †a , a
a contradiction, suppose that at 2 . Then since nt , ant is defined and
ant D .at /n 2 by Lemma 2.18. Also n" 2 N and n" < n, whence an" 2 . But
then a D antCn" D ant an" 2 , a contradiction.
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
1287
L EMMA 4.6. Let a 2 G. / n f1g. Then:
(1) a
1
2 G./ and ŒXa 1  D . 1/ ŒXa ;
(2) b 2 ) bab
1
2 G. / and ŒXbab
(3) ŒXa  D O , a 2
1
 D ŒXa ;
G o . /;
(4) L.G/ D fŒXa  j a 2 G. /g.
Proof. (1) is straightforward. As to (2), let b 2 and let WD ord.a/. We know
that D O./. One can show by internal induction on that .bab 1 / is defined
and equal to ba b 1 for all . Thus, for i D o. /, we have .bab 1 /i 2 and
hence bab 1 2 G./. Again, for r 2 †a , we have that .bab 1 /Œr  is defined and
equals baŒr  b 1 , whence ŒXa  D ŒXbab 1 .
The proof of (3) is immediate from the definitions and Lemma 3.1. Now
suppose X 2 L.G/ and X 2 X. Suppose domain.X / D . r; r/ and consider the
nonstandard extension X W . r; r/ ! G of X. Let a D X. 1 / 2 . We claim that
X D ŒXa . To see this, let s < min.r; ra / and let " be an infinitesimal element of

R such that s D Œs
C ". Then:
1 Œs 
Œs 
D st X
D st aŒs  D Xa .s/:
X.s/ D st X
Hence we have shown that X and Xa agree on . r; r/ \ . ra ; ra /. This proves (4).
The local exponential map. Consider the following sets:
K WD fX 2 L.G/ jI domain.X/ and X.I / Ug
and
K WD fX.1/ j X 2 Kg:
L EMMA 4.7. For every X 2 L.G/, there is s 2 .0; 1/ such that s X 2 K.
Proof. Let X 2 X and suppose X W . r; r/ ! G. If r 1, pick any 0 < s1 < r
and then s1 X W . sr1 ; sr1 / ! G will have I contained in its domain. If r > 1, let
s1 D 1. By continuity of s1 X , there is 0 < s2 < 1 such that .s2 .s1 X //.I / U.
So for s WD s1 s2 , s X 2 K.
L EMMA 4.8. The map X ! X.1/ W K ! K is bijective.
Proof. Let X1 and X2 be in K with representatives X1 and X2 containing I
in their domains. Suppose X1 .1/ D X2 .1/. Then .X1 . 21 //2 D .X2 . 12 //2 , implying
that X1 . 12 / D X2 . 21 / since U is a special neighborhood. Inductively, we have
X1 . 21n / D X2 . 21n / for all n, and thus X1 . 2kn / D X2 . 2kn / for all k 2 Z such that
j 2kn j 1. By density, we have ŒX1  D ŒX2 .
From now on, for X 2 K, we let E.X/ D X.1/. So the last lemma now reads
that E W K ! K is a bijection. We will have much more to say about this local
exponential map later and it will be a crucial tool in our solution of the Local H5.
1288
ISAAC GOLDBRING
A countable neighborhood basis of the identity. For Q as earlier, we put
ord.Q/ WD ordU .Q/. Extend this to the standard setting as follows: for symmetric
P G with 1 2 P , let ord.P / be the largest n such that P n is defined and P n U
if there is such an n and set ord.P / WD 1 if P n is defined and P n U for all n.
We set Vn WD fx 2 G j ord.x/ ng. Notice that for all n,
p1 1 .U/ \ \ pn 1 .U/ Vn Un :
L EMMA 4.9. .Vn W n 1/ is a decreasing sequence of compact symmetric
neighborhoods of 1 in G, ord.Vn / ! 1 as n ! 1, and fVn W n 1g is a countable
neighborhood basis of 1 in G.
Proof. For > N, consider the internal set
V WD fg 2 G j ord.g/ g:
Note that V , so given any neighborhood U of 1 in G, we have that x1 xm
is defined and in U for all m and x1 ; : : : ; xm 2 V . It follows that for any neighborhood U of 1 in G and any m we have that .Vn /m is defined and contained in
U for all sufficiently large n.
5. Local Gleason-Yamabe lemmas
The trickiest and most technical part of the original solution of Hilbert’s fifth
problem concerned several ingenious lemmas proved by Gleason and Yamabe.
These lemmas really drive much of the proof and obtaining local versions of them is
a crucial part of our current project. Fortunately, it is not too difficult to obtain local
versions of these lemmas, although carrying out some of the details can become
laborious.
Throughout this section, we work with the following assumptions. We suppose that U W are compact symmetric neighborhoods of 1 such that W UM and
UP W for fixed positive integers P M large enough for all of the arguments
below to make sense. (A careful analysis of the arguments below could give precise
values for M and P , but this endeavor became quite tedious.) We let Q U be
symmetric such that 1 2 Q, N WD ordU .Q/ 6D 1, and QN C1 is defined. We first
define the map  D Q W G ! Œ0; 1 by
.1/ D 0;
.x/ D
.x/ D 1 if x … QN .
i
N C1
if x 2 Qi n Qi
1,
1 i N;
Then for all x 2 G,
.x/ D 1 if x … U;
j.ax/
.x/j 1
N
for all a 2 Q such that .a; x/ 2 .
1289
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
To smooth out , fix a continuous function W G ! Œ0; 1 such that
.1/ D 1 and .x/ D 0
for all x 2 G n U:
It is important later that depends only on U, not on Q.
Next define D Q W G ! Œ0; 1 by setting
(
supy2U .1 .y// .y 1 x/ if x 2 W
.x/ D
0
if x 2 G n W.
L EMMA 5.1. The following properties hold for the above defined functions:
(1) is continuous and .x/ D 0 if x … U2 ;
(2) 0 1;
(3) j.ax/
.x/j 1
N
for a 2 Q if .a; x/ 2 .
Proof. It is clear that .x/ D 0 if x … U2 . To see that is continuous, let
x 2 G and a 2 . We must show that .ax/ .x/ is infinitesimal. First note
that if x … W, then ax … W ; in this case, .ax/ D .x/ D 0. So assume that
x 2 W. If ax … W, then x … interior.W/. Since U2 interior.W/, we conclude
that x … U2 and so .x/ D .ax/ D 0. Now suppose that ax 2 W , so we can use
the first clause in the definition of for .x/ and .ax/. In this case, .ax/ .x/
is infinitesimal by the continuity of . .2/ is also clear from the definition.
We now prove (3). Let a 2 Q and suppose .a; x/ 2 . If x … W, then ax … U2 ,
else x 2 U3 W, a contradiction. Hence .ax/ D .x/ D 0 in this case. Now
if x 2 W n U3 , then ax … U2 , and once again .ax/ D .x/ D 0. Hence we can
suppose that x 2 U3 so that ax 2 U4 W and so we can use the first clause in the
definition of for .x/ and .ax/. Let y 2 U. Then .a 1 ; y/ 2  and
j.1
Since y
1 ax
j.1
.a
1
y//
is defined, we have y
.y// .y
1
ax/
.1
1 ax
.1
.y//j D .a
.a
1
1
:
N
1 y/ 1 x,
y/ ..a
1
so
y/
1
x//j 1
:
N
Now, (3) follows from noting that the sets f.1 .y// .y 1 x/ j y 2 Ug and
f.1 .a 1 y//..a 1 y/ 1 x/ j y 2 U; a 1 y 2 Ug have the same supremum. Let C WD ff W G ! R j f is continuous and supp.f / W2 g: Then C is a
real vector space and we equip it with the norm given by
kf k WD supfjf .x/j j x 2 Gg:
If f 2 C is such that supp.f / W and a 2 W, then we can define the new
function a f by the rule
(
f .a 1 x/ if x 2 W2
.a f /.x/ D
0
otherwise:
1290
ISAAC GOLDBRING
It is easy to see that a f 2 C and that if f; g 2 C both have supports contained
in W, then a .f C g/ D a f C a g. We will often drop the and just write
af instead of a f . This should not be confused with the function rf given by
.rf /.x/ D rf .x/ for r 2 R, which is defined for any f W G ! R.
L EMMA 5.2. Suppose a; b 2 W and f 2 C is such that supp.f / W and
supp.bf / W. Then:
(i) kaf k D kf k;
(ii) a.bf / D .ab/f and k.ab/f
f k kaf
f k C kbf
f k.
Proof. To see (i), note that
kaf k D supfjf .a
1
x/j j x 2 W2 g
D supfjf .a
1
x/j j x 2 W2 ; a
1
x 2 Wg
D supfjf .y/j j y 2 Wg
D kf k:
We now prove (ii). Suppose x … W2 . In this case, ..ab/f /.x/ D 0 and since
a
and supp.bf / W, we also have .a.bf //.x/ D 0. Now suppose x 2 W2 .
Then ..ab/f /.x/ D f ..ab/ 1 x/ D f ..b 1 a 1 /x/ and .a.bf //.x/ D .bf /.a 1 x/.
If a 1 x … W2 , then b 1 a 1 x … W; in this case, we have ..ab/f /.x/ D 0 and
.a.bf /.x/ D 0. If a 1 x 2 W2 , then
1x … W
.a.bf //.x/ D .bf /.a
1
x/ D f .b
1
.a
1
x// D ..ab/f /.x/:
We now finish the proof of (ii) as in the global case, that is
k.ab/f
f k D ka.bf /
ka.bf /
ka.bf
D kbf
af C af
fk
af k C kaf
f /k C kaf
f k C kaf
fk
fk
f k:
Since supp./ U2 W, we can consider the function a 2 C for any a 2 W.
We will often need the following equicontinuity result.
(4) For each " 2 R>0 , there is a symmetric neighborhood V" of 1 in G, independent
of Q, such that V" U and ka k " for all a 2 V" .
To see this, let " 2 R>0 and note that by the uniform continuity of , we have
that a neighborhood U of 1 in G such that j .g/ .h/j < " for all g; h 2 U such
that gh 1 2 U . Now take a symmetric neighborhood V" of 1 in G such that V" U
and y 1 ay 2 U for all .a; y/ 2 U V" . The claim is that this choice for V" works.
Fix a 2 V" . We must compute j.a /.x/ .x/j for various x. Let us first take
care of some of the trivial computations. If x … W2 , then .a /.x/ D .x/ D 0.
Next suppose that x 2 W2 n W, so .x/ D 0. If a 1 x … W, then .a /.x/ D 0.
If a 1 x 2 W, then a 1 x … U2 , else x 2 U3 W, a contradiction. So in this
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
1291
case also, .a/.x/ D 0. Now if x 2 W n U3 , then a 1 x … U2 , so once again
.a /.x/ D .x/ D 0. Thus the only real case of interest is when x 2 U3 and hence
a 1 x 2 U4 W. In this case, we have j .y 1 a 1 x/ .y 1 x/j < " for any y 2 U.
We now have
ˇ
ˇ
j.a /.x/ .x/j D ˇ sup .1 .y// .y 1 a 1 x/ sup .1 .y// .y 1 x/ˇ
y2U
sup .1
y2U
.y//j .y
1
a
1
x/
.y
1
x/j
y2U
":
This finishes the proof of (4).
In [12], Montgomery and Zippin construct a linear functional
Z
f 7! f W C ! R
with the following properties:
R
R
(i) For any f 2 C , one has j f j jf j;
(ii) (Left
invariance)
Suppose f 2 C satisfies supp.f / W. Let a 2 W. Then
R
R
af D f .
The above defined functional will be referred to as the local Haar integral
on G. As is the case for topological groups, the local Haar integral is unique up to
multiplication by a positive real constant. By the Riesz Representation Theorem,
for each local Haar integral as above, we get a Borel measure on a certain algebra of subsets of W2 . We will refer to the measure corresponding to a local
Haar integral as a local Haar measure on G. If a given integral
induces the measure
R
, then we will denote the
integral
of
a
function
f
by
or if we wish to
f
d,
R
specify the variable, by f .x/d.x/. By the left invariance of the integral, the
measure is left-invariant in the sense that for any measurable subset V W and
any a 2 W, we have that .aV / D .V /.
With these remarks behind us, we can now proceed with our discussion. Fix
a continuous function 1 W G ! Œ0; 1 such that
1 .x/ D 1 for all x 2 U2 ;
1 .x/ D 0 for all x 2 G n U3 :
It is important later that 1 depends only on U, not on Q.
Note that 0 1 . Take the unique local Haar measure such that
R
1 .x/d.x/ D 1. Then in particular,
R
(5) 0 .x/d.x/ 1:
By (4) above, we have an open neighborhood V U of 1 in G, independent
of Q, such that 2 .x/ 21 on V . We now introduce the function
D Q W G ! R
1292
ISAAC GOLDBRING
defined by
(R
.x/ D
.xu/.u/d.u/ if x 2 W
if x 2 G n W.
0
It should be clear that is continuous. The following are other useful properties of :
(6) supp./ U4 W;
(7) .1/ .V /
2 ;
(8) if a 2 Q, then ka
(9) ka
k ka
k 1
N
;
k for all a 2 W.
In verifying properties (8) and (9), one needs to perform the case distinctions
that we have done earlier. As before, though, there is only one case where the
first clause of the definition applies for both a and and in this case the results
easily follow from the properties of the local Haar integral and the earlier derived
properties (1)–(5). We use that U6 W in showing that there is only one genuine
calculation to perform.
We should also point out that, by (6), supp.a / U5 W for all a 2 W,
and so for all b 2 W, we can consider the function b.a /. We are finally in
the position to prove the local versions of the Gleason-Yamabe lemmas.
L EMMA 5.3. Let c; " 2 R>0 . Then there is a neighborhood U D Uc;" U of
1 in G, independent of Q, such that for all a 2 Q, b 2 U , and m cN ,
kb m.a
/
/k ";
m.a
km.a
/k c:
Proof. Let a 2 Q and b 2 U. Note that supp.b m.a / m.a // U6 .
So let x 2 U6 and set y WD b 1 x. Then
Z
.a /.x/ D Œ.a 1 xu/ .xu/.u/ d.u/
Z
b.a /.x/ D .a /.y/ D Œ.a 1 yu/ .yu/.u/ d.u/:
Assuming P is large enough so that y 1 x 2 W, we can use left-invariance
of the integral to replace u by x 1 yu in the function of u integrated in the first
identity. Thus,
Z
.a /.x/ D Œ.a 1 yu/ .yu/.x 1 yu/d.u/:
Taking differences gives
Z
Œb .a
/
.a
/.x/ D
Œ.a
/.yu/Œ.
y
1
x /.u/d.u/:
1293
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
By (4), we can take the neighborhood Uc;" U of 1 in G so small that for all
b 2 Uc;" and x 2 U6 we have that y 1 x 2 U and
"
k y 1 xk <
:
c.U3 /
It is straightforward to see that this choice of Uc;" works.
L EMMA 5.4. With c; " 2 R>0 , let U D Uc;" be as in the previous lemma and
let a 2 Q and m and n be such that m cN , n > 0, an is defined, and ai 2 U for
n
i 2 f0; : : : ; ng. Then k. m
/ m.a /k ".
n /.a P
Proof. It is routine to check that m.an / D ni D01 ai m.a /. Hence,
m.an /
nm.a
/ D
n
X1
.ai m.a
/
m.a
//:
i D0
By the previous lemma, we have for i D 0; : : : ; n
i
ka m.a
/
m.a
1,
/k ";
which gives the desired result by summing and dividing by n.
Now suppose that Q is a symmetric internal subset of with 1 2 Q such that
N WD ordU .Q/ 2 N . Note that N > N. Then everything we have done so far
in this section transfers to the nonstandard setting and yields internally continuous
functions W G ! Œ0; 1 and W G ! R satisfying the internal versions of
(1)–(9) and Lemmas 5.3 and 5.4. We now have the following lemma.
L EMMA 5.5. Suppose a 2 Q, D O.N /, a is defined, and a 2 for all
. Then k.a /k is infinitesimal.
Proof. Let " 2 R>0 . Choose c 2 R>0 such that cN . Then by Lemma 5.4,
with m D n D ,
k.a / .a /k ":
Also ka k ka k < " by (9) and (4), so k.a
/k 2":
6. Consequences of the Gleason-Yamabe lemmas
In this section, we bear the fruits of our labors from the previous section. We
derive some consequences of the Gleason-Yamabe lemmas which allow us to put
a group structure on L.G/. From this group structure on L.G/, we obtain the
important corollary that in an NSS local group, the image of the local exponential
map is a neighborhood of 1, which later will allow us to conclude that NSS local
groups are locally euclidean.
Group structure on L.G/.
L EMMA 6.1. Suppose 2 N n N and a1 ; : : : ; a is a hyperfinite sequence
such that ai 2 G o ./ for all i 2 f1; : : : ; g. Let Q WD f1; a1 ; : : : ; a ; a1 1 ; : : : ; a 1 g.
Then Q is defined and Q .
1294
ISAAC GOLDBRING
Proof. We first show that if Q is defined, then Q . Suppose, towards
a contradiction, that Q ª . Take a compact symmetric neighborhood U of 1 in
G such that QC1 ª U , so ordU .Q/ . Fix W as in the setting of the GleasonYamabe lemmas and choose U small enough so that U W and U P W. By
decreasing if necessary, and Q accordingly, we arrange that ordU .Q/ D .
Consider first the special case that Qi for all i D o./. Take b 2 Q
such that st.b/ 6D 1. Choose U as in the setting of the Gleason-Yamabe lemmas so
that U U and st.b/ … U4 . Let WD ordU .Q/ and note that D O./, else we
would have that Q , a contradiction. By the Transfer Principle, the previous
section yields the internally continuous function D Q W G ! R satisfying the
internal versions of all of the properties and lemmas from that section. In particular,
.b 1 / D 0 and .1/ is not infinitesimal. Hence kb k is not infinitesimal.
Take a hyperfinite sequence b1 ; : : : ; b from Q such that b D b1 b . Then by
the Transfer Principle and Lemma 5.2,
kb
k X
kbi k
i D1
and the right-hand side of the inequality is infinitesimal by Lemma 5.5, yielding a
contradiction.
Now assume that Qi ª for some i D o./. Then the compact subgroup
GU .Q/ of G as defined in Lemma 3.9 is nontrivial (and contained in U ). By
a standard consequence of the Peter-Weyl Theorem (see [12, p. 100]), we can
find a proper closed normal subgroup K of GU .Q/ and a compact symmetric
neighborhood V U of 1 in G such that K interior.V / and every subgroup
of GU .Q/ contained in V is contained in K. Put WD ordV .Q/, so 2 N n N
and . Since the compact subgroup GV .Q/ is contained in V , we must have
that GV .Q/ K. Take b 2 Q such that st.b/ … interior.V /. Then st.b/ … K,
so we can take a compact symmetric neighborhood U of 1 as in the setting of the
Gleason-Yamabe lemmas such that K interior.U/, U V , and st.b/ … U4 . Put
WD ordU .Q/. If D o./, then st.Q / GV .Q/ K, contradicting the fact
that st.Q / ª interior.U/. Hence D O./. The rest of the proof now proceeds
as in the special case considered earlier, with replaced by , and b1 ; : : : ; b by
an internal sequence b1 ; : : : ; b in Q such that b D b1 b .
We now prove that Q is defined. To do this, we first fix an internal set E .
j
C LAIM. Let c1 ; : : : ; c be an internal sequence such that ci is defined and
j
ci 2 E for all i; j 2 f1; : : : ; g. Let R D f1; c1 ; : : : ; c ; c1 1 ; : : : ; c 1 g. Then R
is defined.
We prove this claim by internal induction on . The claim is clearly true
for D 1 and we now suppose inductively that it holds for a given . Suppose
j
j
c1 ; : : : ; cC1 is an internal sequence such that ci is defined and ci 2 E for all i; j 2
1
f1; : : : ; C 1g. Let R D f1; c1 ; : : : ; cC1 ; c1 1 ; : : : ; cC1
g. Fix d1 ; : : : ; dC1 2 R.
1295
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
By the induction hypothesis and the first part of the proof, d1 d is defined and
infinitesimal; similarly, d1 di and di dC1 are defined and infinitesimal for
all i 2 f2; : : : ; C 1g. By Lemma 2.18, d1 dC1 is defined. Since d1 ; : : : ; dC1
from R were arbitrary, we have that RC1 is defined. This finishes the proof of
the claim.
Now given an internal sequence a1 ; : : : ; a as in the statement of the lemma,
j
the claim implies that Q is defined by taking E to be the set of all ai for i; j 2
f1; : : : ; g.
Remark 6.2. In the global version of this lemma appearing in [4], it was required that G was pure. Van den Dries was able to remove this assumption and in
the proof of the previous lemma we have adapted his method to the local setting.
L EMMA 6.3. Suppose that U is a compact symmetric neighborhood of 1 in G
with U U2 . Let > N be such that for all i 2 f1; : : : ; g, ai and b i are defined
and ai 2 U , b i 2 . Then for all i 2 f1; : : : ; g, we have that .ab/i is defined and
.ab/i ai .
Proof. Suppose i 2 f1; : : : ; g. Then .ai ; b/ 2  and ai b ai . Since
2 , we have .ai b; a i / 2  . Similarly, .ai ; ba i / 2  . Thus
we can define the element bi WD ai ba i . For similar reasons, for any i; j 2
f1; : : : ; g, we can define the element ai b j a i . Note that bi 2 for all i 2
f1; : : : ; g.
.st.ai /; st.a i //
j
j
C LAIM 1. For all i; j 2 f1; : : : ; g, bi is defined and bi D ai b j a i .
We prove Claim 1 by internal induction on j . The case j D 1 is obvious.
Suppose that the assertion is true for all j 0 2 f1; : : : ; j g and further suppose that
j C1
j C 1 . We first show that bi
is defined. Let k; l 2 f1; : : : ; j g be such
that k C l D j C 1. By the induction hypothesis, we know that bik is defined and
bik D ai b k a i . Since b k 2 , we have that bik 2 . Similarly, we have that bil is
j C1
defined and bil 2 ; thus .bik ; bil / 2  . So by Lemma 2.18, we have that bi
is
j
defined. But now, using that bi 2 ,
j C1
bi
j
D bi bi
D .ai b j a i /.ai ba i /
D ..ai b j a i / .ai // .ba i /
D .ai b j / .ba i /
D ..ai b j / b/a
i
D ai b j C1 a i :
j
This proves Claim 1. Note that we have also shown that bi 2 for all i; j 2
f1; : : : ; g. Then by Lemma 6.1, b1 bi is defined and b1 bi 2 for all i 2
f1; : : : ; g.
1296
ISAAC GOLDBRING
The following claim finishes the proof of the lemma.
C LAIM 2. For all i 2 f1; : : : ; g, .ab/i is defined and .ab/i D .b1 bi /ai .
We prove Claim 2 by internal induction on i . The assertion is clearly true for
i D 1 and now suppose that it holds for all j 2 N with j i . Suppose i C 1 .
To prove that .ab/i C1 is defined, it is enough to check that ..ab/k ; .ab/l / 2 
for all k; l 2 f1; : : : ; i g with k C l D i C 1. By the induction hypothesis, .ab/k D
.b1 bk /ak ak and .ab/l D .b1 bl /al al and the desired result now follows
from the fact that .ak ; al / 2 U U . Next note that
ai C1 b D .ai C1 b/ .a
i 1 i C1
a
/ D .ai C1 ba
i 1
/ai C1 D bi C1 ai C1 :
Hence, by the induction hypothesis, we have
.ab/i C1 D ..ab/i / .ab/
D ..b1 bi /ai / .ab/
D .b1 bi / .ai .ab//
D .b1 bi /.ai C1 b/
D .b1 bi /.bi C1 ai C1 /
D ..b1 bi /bi C1 /aiC1
D .b1 bi C1 /ai C1 :
L EMMA 6.4. Let > N and a 2 G./ be such that a is defined and ai 2 Gns
for all i 2 f1; : : : ; g. Suppose also that b 2 is such that b is defined and b i 2 Gns
i
i
1
o
and a b for all i 2 f1; : : : ; g. Then a b 2 G ./.
Proof. We first make a series of reductions. Note that by Lemma 6.3, we have
that a 1 b 2 G./. Let V be a compact symmetric neighborhood of 1 in G such
that V U2 . Then since D O.ordV .a 1 b//, by replacing with ordV .a 1 b/ if
> ordV .a 1 b/, we may as well assume that .a 1 b/i is defined and .a 1 b/i 2 Gns
1
1
for all i 2 f1; : : : ; g. Let Q D f1; a; a ; b; b g. Suppose that Q is not defined.
Let 2 N n N be the largest element of N such that Q is defined. If D o./,
then by Lemma 6.1, Qi for all i 2 f1; : : : ; g. But then, by Lemma 2.18,
QC1 is defined, a contradiction. Thus D O./. So, by replacing by if
necessary, we can assume that Q is defined. If a 2 G o ./, then by Lemma 6.3,
.a 1 b/i is defined and .a 1 b/i a i 2 for all i 2 f1; : : : ; g, yielding the
desired conclusion. So we may as well assume that a … G o ./ and by replacing by a smaller element of its archimedean class, we may as well assume that a … .
Now suppose, towards a contradiction, that there is j 2 f1; : : : ; g such that
1
.a b/j … . Note that we must have D O.j /. Choose compact symmetric
neighborhoods U and W of 1 in G as in the previous section so that
a 2 W n U and .a
1
b/j 2 W n .U /4 :
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
1297
Let D ordU .Q/. Then D O. / by Lemma 6.1. Let D Q W G ! R be the
internally continuous function constructed in the previous section. Then we know
that ..a 1 b/j / D 0 and " WD .1/ > 0 and is not infinitesimal. Also, note that
" k.b
1
j k.b
a/j k
1
k
a/
D j ka
bk
D kj.a
/
j.b
/k:
We will obtain a contradiction by showing that kj.a / j.b /k < ": By
Lemma 5.4, there is a compact symmetric neighborhood U U of 1 in G such
that for all k > 0 in N , if ai 2 U and b i 2 U for all i 2 f1; : : : ; kg, then
j j "
"
.ak / j.a / < ;
.b k / j.b / < :
3
3
k
k
Let k D minfordU .a/; ordU .b/g. Then the above equalities hold for this k.
Note that D O.k/ and hence j D O.k/. Since k < , we have ak b k and so
j
j j .ak /
.b k / D kak b k k 0:
k
k
k
Combining this fact with the previous inequalities yields
kj.a
/
j.b
/k < ";
giving the desired contradiction.
T HEOREM 6.5. Suppose > N. Then
(1) G./ and G o ./ are normal subgroups of ;
(2) if a 2 G./ and b 2 , then aba
1b 1
2 G o . /;
(3) G./=G o ./ is abelian.
Proof. We have already remarked that G. / and G o . / are symmetric and
that G./ is closed under multiplication. That G./ is a normal subgroup of is
part of the content of Lemma 4.6, (2). Lemma 6.3 implies that G o . / is a group.
Now suppose that a 2 G o . / and b 2 . We need bab 1 2 G o . /. One can show
by internal induction on that .bab 1 / is defined and equal to ba b 1 . In
particular, this shows that bab 1 2 G o . /. This finishes the proof of (1).
Let a 2 G./ and b 2 . We need to show that aba 1 b 1 2 G o . /. Note
that ba 1 b 1 2 G./ by (1). Let U U2 be a compact symmetric neighborhood
of 1. Choose 2 f1; : : : ; g such that D O. /, a and .ba 1 b 1 / are defined,
and ai 2 U for i 2 f1; : : : ; g. One can prove by internal induction that for
i 2 f1; : : : ; g, .ba 1 b 1 /i D ba i b 1 and hence a i .ba 1 b 1 /i . By Lemma
6.4, aba 1 b 1 2 G o . / G o . /. This finishes (2) and (2) obviously implies (3).
1298
ISAAC GOLDBRING
Observe that if X 2 L.G/ and > N, then X. 1 / 2 G. /. To see this, suppose
i
that i D o./. Then X. 1 / D X. i / 2 .
T HEOREM 6.6. The map S W L.G/ ! G. /=G o . / defined by
1
G o . /
S.X/ D X
is a bijection.
Proof. Suppose that X; Y 2 L.G/ and S.X/ D S.Y/. Letting a WD X. 1 /
and b WD Y. 1 /, we have that a 1 b 2 G o . /. Let U be a compact symmetric
neighborhood of 1 in G with U U2 . Let WD minfordU .a/; g. By Lemma 6.3,
we have that .a .a 1 b//i is defined, nearstandard, and infinitely close to ai for all
i 2 f1; : : : ; g, i.e., ai b i for all i 2 f1; : : : ; g. Then for all i 2 f1; : : : ; g such
that i 2 domain.X/ \ domain.Y/, we have X. i / D Y. i /. Since D O. /, this
implies that X D Y. Hence S is 1-1.
Now suppose b 2 G. / n G o . /. Consider the local 1-ps Xb of G defined
by Xb .t/ D st.b Œt  / on its domain; see Lemma 3.14. Let b1 D Xb . 1 / 2 .
Choose 2 f1; : : : ; g such that D O. / and 2 domain.Xb /. Note then that
b1 is defined. For any i 2 f1; : : : ; g, b i is defined and b i b1i . By Lemma 6.4,
b 1 b1 2 G o ./ G o . / and hence bG o . / D b1 G o . /: Hence S is onto.
We use Theorem 6.6 to equip L.G/ with an abelian group operation C . More
explicitly, for X; Y 2 L.G/, we define X C Y WD S 1 .S.X/ S.Y//. By the proof
of Theorem 6.6, we see that if t 2 domain.X C Y/ and is such that t, then
h i
1
1
.X C Y/.t / X
Y
:
In [4], a proof is given, in the global setting, that this group operation is
independent of . A local version of that proof is probably possible, but this independence also follows from the Local H5, which we prove without using that C
is independent of . In light of this independence of , we write X C Y instead of
X C Y.
Existence of square roots. In the rest of this section, we assume that G is NSS
and that U is a special neighborhood of 1 in G. We now lead up to the proof of the
crucial fact that the image of the local exponential map is a neighborhood of 1 in
G. In particular, we will see that there is an open neighborhood of 1 in G which is
ruled by local 1-parameter subgroups in the sense that every element in this open
neighborhood lies on some local 1-ps of G. In the rest of this section, we do not
follow [4], but rather [17], which contains the following series of lemmas in the
global setting.
L EMMA 6.7. Suppose > N and a; b 2 G. /. Then ŒXa  C ŒXb  D ŒXab .
1299
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
Proof. From Theorem 6.6,
S.ŒXa  C ŒXb / D S.ŒXa / S.ŒXb /
D .aG o . //.bG o . //
D .ab/G o . /
D S.ŒXab /:
The result now follows since S is injective.
The global version of the next lemma is in Singer [16], but the proof there is
unclear.
L EMMA 6.8. Suppose > N and a 2 G. /. Then a D bc 2 with b 2 G o . /
and c 2 G./.
1
Proof. Let c WD Xa . 2
/ 2 . Then c 2 D Xa .
that ŒXc 2  D ŒXa 1 . From this, we conclude that
ŒXac
Let b WD ac
2.
2
 D ŒXa  C ŒXc
2
1
/ D Xa
1
. 1 /, which implies
 D ŒX1  D O:
Then since ŒXb  D O, we have b 2 G o . /.
L EMMA 6.9. There exists q 2 Q>0 such that for all a; b; c 2 , if a D bc 2
and ord.b/ ord.a/, then ord.c/ q ord.a/.
Proof. Suppose no such q exists. Then for all n 1, there are an ; bn ; cn 2 such that an D bn cn2 with ord.bn / ord.an / and n ord.cn / < ord.an /. By saturation,
we have a; b; c 2 such that a D bc 2 , ord.b/ ord.a/, and n ord.c/ < ord.a/ for
all n. Let WD ord.c/ > N. Then
O D ŒXa  D ŒXbc 2  D ŒXb  C ŒXc 2  D O C 2ŒXc ;
implying that ŒXc  D O, an obvious contradiction.
Fix q as in the above lemma. For a; b 2 , we let Œa; b WD aba 1 b
commutator of a and b. We will frequently use the identity ab D Œa; bba.
1,
the
L EMMA 6.10. Let a 2 . Then for all 1, there are b ; c 2 G such that
b c c is defined, a D b c2 , ord.b / ord.a/ and ord.c / q ord.a/.
Proof. Note that the assertion is trivial if a D 1 and so we suppose that a 6D 1.
Next note that if the assertion is true for a given , then we have that b ; c 2 as they will have infinite order. By Lemmas 6.8 and 6.9, the assertion is true for
some > N. It is clearly then true for every 0 2 f1; : : : ; g. Hence, if we can
prove the assertion for C 1, then we will be done by internal induction. If b D 1,
then the assumed factorization witnesses the truth of the assertion for all 0 . We
thus assume that b 6D 1. By Lemma 6.8, there are b; c 2 and 1 > N such
that b D bc 2 with ord.b/ 1 ord.b /. Note that then ord.b/ 1 ord.a/. By
1300
ISAAC GOLDBRING
Lemma 6.9, we also know that ord.c/ q ord.b / q ord.a/. Since is a group,
a D bc 2 c2 D bccc c
D bcŒc; c c cc
D bŒc; Œc; c Œc; c cc cc
D bŒc; Œc; c Œc; c .cc /2
D bC1 .cC1 /2 ;
where bC1 WD bŒc; Œc; c Œc; c  and cC1 WD cc . Now note that there is 2 > N
such that
ord.Œc; c / 2 ord.c/ 2 q ord.a/:
Also there is 3 > N such that
ord.Œc; Œc; c / 3 2 ord.c/ 3 2 q ord.a/:
In combination with the facts that ord.b/ 1 ord.a/ and G. ord.a// is a group,
where WD min.1 ; 2 /, we have that ord.bC1 / . C 1/ ord.a/. Finally, Lemma
6.9 yields that ord.cC1 / q ord.a/ and so we are finished.
L EMMA 6.11. For every a 2 , there is b 2 with a D b 2 .
Proof. Fix > N. Then by the previous lemma, the internal set of all 2 N
such that for each x 2 V there are y; z 2 U such that y z z is defined, x D yz 2 ,
ord.y/ ord.x/, and ord.z/ q ord.x/ includes all infinite . By underspill,
there is some n > 0 in this set. Now let x 2 Vn . Then there are y; z 2 U such that
y z z is defined, x D yz 2 , and ord.y/ ord.x/. Since ord.y/ is infinite, y 2 .
Hence x D st.z/2 , i.e., every element of Vn has a square root in U. By transfer,
this implies that every a 2 has a square root b 2 U and b 2 , else the group
f1; st.b/g is contained in U, a contradiction.
L EMMA 6.12. Suppose that a 2 , b 2 G , and > 0 are such that ord.b/ and a D b . Then for all i 2 f1; : : : ; g, we have that b i 2 .
Proof. We begin the proof with a claim.
C LAIM. For all n, b n is defined.
We prove this claim by induction on n, the case n D 1 being true by assumption.
Assume inductively that b n is defined. We claim that
.Ž/
b j 2 U2 for all j 2 f1; : : : ; ng:
To see this, write j D n0 C j 0 , where n0 2 N and j 0 2 f1; : : : ; j
n0
b D .b / b
j0
b
j0
1g. Then
2U :
In order to show that b nC is defined, we prove by internal induction on i that
b nCi is defined and b nCi 2 U2 for all i 2 f0; : : : ; g. The case i D 0 is true since
b n D an 2 . Suppose, inductively, that it holds for a given i and that i C 1 .
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
1301
Let k; l 2 f1; : : : ; n C ig be such that k C l D n C i C 1. Then by .Ž/ and the
inductive assumption, .b k ; b i / 2 U2 U2  . Hence b nCi C1 is defined. But
then b i C1 2 U and b nCi C1 b i C1 . Hence b nCi C1 2 U2 . This proves the claim.
Now suppose, towards a contradiction, that there is i 2 f1; : : : ; 1g such
that b i … . Let x WD st.b i / 2 U n f1g. From the claim, we see that if x n is defined
for a given n, then x n 2 U. From this, one can show by induction that for all n, x n
is defined and x n 2 U. Hence x Z U, a contradiction.
L EMMA 6.13. Given a 2 and , there is b 2 such that ord.b/ 2 and
a D b2 .
Proof. We prove this by internal induction on , noting that the assertion
is internal since b 2 can be replaced by b 2 G by the previous lemma. The
assertion is true for D 1 by Lemma 6.11. Now suppose that the assertion is true
for , i.e., that a D b 2 with ord.b/ 2 . Since b 2 , there is c 2 , with b D c 2
by Lemma 6.11.
C LAIM. For all i 2 f1; : : : ; 2 g, c 2i is defined and .c 2 /i D c 2i .
We prove the claim by internal induction. It is obviously true for i D 1 and
suppose it is true for a given i 2 . Suppose i C 1 2 . We first show c 2i C1 is
defined. By induction, we know that c 2i is defined, and so by Lemma 2.18, it is
enough to show .c k ; c l / 2  for k; l 2 f1; : : : ; 2i g with k C l D 2i C 1. If k is
even, then
k
k
c k D .c 2 / 2 D b 2 2 by Lemma 6.12 since
k
2
i 2 . Similarly, if k is odd, then
c k D c .c 2 /
k 1
2
D cb
k 1
2
2 :
.c k ; c l /
Thus, for k; l 2 f1; : : : ; 2i g, one has
2  . Now one can repeat
the same argument using that c 2iC1 is defined to conclude that c 2i C2 is defined.
Now that we know that c 2iC2 is defined, we have, by induction, that
.c 2 /i C1 D .c 2 /i c 2 D c 2i c 2 D c 2iC2 :
C1
By the claim, we have that a D .c 2 /2 D c 2 . It remains to show that
i
ord.c/ 2C1 . Let i 2 f1; : : : ; 2C1 g. Then if i is even, we have that c i D b 2 2 i
1
by Lemma 6.12 since 2i 2 ord.b/. If i is odd, then c i D c c 2 2 for the
same reason. Thus c i 2 for all i 2 f1; : : : ; 2C1 g and so ord.c/ 2C1 .
T HEOREM 6.14. K is a neighborhood of 1.
Proof. Fix > N. Then the internal set of all such that for each x 2 V
there is y 2 U such that ord.y/ 2 and x D y 2 contains all infinite . So by
underspill, there is some n > 0 in this set. Now given x 2 Vn , there is a 2 U such
that ord.a/ 2 and x D a2 . Let WD 2 . Then for t 2 I , Xa .t / D st.aŒt  / 2 U
and Xa .1/ D x. Thus ŒXa  2 K and x D ŒXa .1/ 2 K. We have shown Vn K
and this completes the proof.
1302
ISAAC GOLDBRING
7. The space L.G/
Throughout this section, we assume that our (locally compact) G is NSS. We
fix a special neighborhood U of 1 in G. We will equip L.G/ with a topology in
such a way that it becomes a locally compact finite dimensional real topological
vector space, where the operations of scalar multiplication and vector addition are
as previously defined. The section culminates with the important fact that locally
compact NSS local groups are locally euclidean.
The compact-open topology. We first recall that if X and Y are Hausdorff
spaces, we can turn C.X; Y /, the set of continuous maps from X to Y , into a
topological space by equipping it with the compact-open topology. This topology
has as a subbasis the sets B.C; U / WD ff 2 C.X; Y / j f .C / U g, where C ranges
over all compact subsets of X and U ranges over all open subsets of Y . In the
solution of the H5, one equipped the space C.R; G/ with the compact-open topology and then gave the space of 1-parameter subgroups the corresponding subspace
topology.
Analogously, we will define a topology on L.G/ by declaring the subbasic
open set to be the sets BC;U WD fX 2 L.G/ j C domain.X/ and X.C / U g,
where C ranges over all compact subsets of . 2; 2/ and U ranges over all open
subsets of G. We next describe the monad structure of L.G/ with respect to this
topology.
L EMMA 7.1. Let X 2 L.G/ and Y 2 L.G/ . Then Y 2 .X/ if and only if
domain.X/ \ . 2; 2/ domain.Y/ and for every t 2 domain.X/ \ . 2; 2/ and
every t 0 2 .t/, Y.t 0 / 2 .X.t //.
Proof. First suppose that Y 2 .X/. Since Y 2 BC;G
for every compact
C domain.X/ \ . 2; 2/, we have that domain.X/ \ . 2; 2/ domain.Y/. Now
suppose that t 2 domain.X/ \ . 2; 2/ and U is an open neighborhood of X.t / in G.
Let C be a compact neighborhood of t with C domain.X/ \ . 2; 2/ and such
that X.C / U . Since Y 2 BC;U
, we have that Y.C / U and so in particular
Y.t 0 / 2 U . Thus Y.t 0 / 2 .X.t //.
For the converse, suppose that domain.X/ \ . 2; 2/ domain.Y/ and for all
t 2 domain.X/ \ . 2; 2/ and t 0 2 .t /, we have that Y.t 0 / 2 .X.t //. Suppose
X 2 BC;U . Then for all t 2 C , we have that Y.t / 2 .X.st.t /// U . This
implies that Y 2 BC;U
and thus Y 2 .X/.
In the rest of the paper, we assume that the special neighborhood U of an NSS
local group is chosen so small that U U6 .
L EMMA 7.2. K is a compact neighborhood of O in L.G/.
Proof. To see that K is a neighborhood of O, choose an open W U with
1 2 W . Let KW WD fX 2 L.G/ j I domain.X/ and X.I / W g K. Clearly
O 2 KW and KW is an open set in L.G/ by the definition of the topology.
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
1303
We now prove that K is compact. Let Y 2 K . It suffices to find X 2 L.G/
such that Y 2 .X/. Choose Y 2 Y with I domain.Y /. Now define X W I ! G
by X.t / D st.Y.t//. By Lemma 3.15, we may extend the domain of X to ensure
that it is a local 1-ps. We claim that X WD ŒX  is such that Y 2 .X/. We first
note that . 2; 2/ domain.Y/. To see this, suppose that t 2 .1; 2/ . Then
.Y .t 1/; Y.1// 2 U U  , so one can define Y .t / WD Y .t 1/Y .1/. One
defines Y.t/ for t 2 . 2; 1/ in a similar fashion. An easy check verifies that
Y W . 2; 2/ ! G is an internal local 1-parameter subgroup. Furthermore, if
t 2 domain.X/ \ .1; 2/ and t 0 2 .t /,
Y.t 0 / D Y.t 0
1/Y.1/ 2 .X.t
1/X.1// D .X.t //:
A similar argument deals with the case that t 2 domain.X/ \ . 2; 1/, finishing
the proof of the lemma.
C OROLLARY 7.3. E W K ! K is a homeomorphism.
Proof. By Lemma 7.1, it is clear that E is continuous. The corollary is now
immediate from the fact that E is a continuous bijection and K is compact.
Suppose that X 2 L.G/ and s 2 .0; 2/ is such that Œ s; s domain.X/ and
X.Œ s; s/ U2 . Let U U2 be a neighborhood of 1. Consider the set
NX .s; U / WD fY 2 L.G/ j Y.t / 2 X.t /U for all t 2 Œ s; sg:
L EMMA 7.4. For each neighborhood U of 1 in G with U U2 , NX .s; U / is
a neighborhood of X in L.G/ and the collection
fNX .s; U / j U is a neighborhood of 1 in G and U U2 g
is a neighborhood base of X in L.G/.
Proof. Let Y 2 .X/. Since Œ s; s domain.X/ \ . 2; 2/ domain.Y/, we
have Œ s; s domain.Y/. Now fix t 2 Œ s; s and U U2 a neighborhood of 1
in G. Then Y.t/; X.t/ 2 .X.st.t //, whence X.t / 1 Y.t / 2 . Thus Y.t / 2 X.t /U and hence Y 2 NX .s; U / . This implies that .X/ NX .s; U / , whence NX .s; U /
is a neighborhood of X.
Now suppose that Y 2 NX .s; U / for every neighborhood U of 1 in G with
U U2 . We must show that Y 2 .X/. We first need to check that domain.X/ \
. 2; 2/ domain.Y/. Say domain.X/ \ . 2; 2/ D . r; r/. Choose n > 0 such
that n1 . r; r/ Œ s; s. Since Y 2 NX .s; U2n / , we see that the domain of Y
contains . r; r/ since the rule Y.t / D Y. nt /n for t 2 . r; r/ defines an internal
local 1-parameter subgroup. Now fix t 2 . r; r/ and t 0 2 .t /. We must show that
Y.t 0 / 2 .X.t//. Let U be a neighborhood of X.t /. We will show that Y.t 0 / 2 U .
Since X. nt / 2 domain.pn /, there is an open neighborhood W1 of X. nt / such that
W1 domain.pn / and pn .W1 / U . By local compactness, we can choose a
compact neighborhood W of 1 in G with W U2 such that fX. nt /g W 
0
0
and X. nt /W W1 . Since Y 2 NX .s; W / , we know that Y. tn / 2 X. tn /W W1 .
1304
ISAAC GOLDBRING
0
0
0
Indeed, suppose a 2 W . Then X. tn /a D X. nt / .X. t nt /a/ and X. t nt /a a,
0
0
whence st.X. t nt /a/ 2 W . Since W1 is open, we see that X. tn /a 2 W1 . We now
0
0
see that Y. tn / 2 domain.pn / and Y.t 0 / D Y. tn /n 2 U .
T HEOREM 7.5. L.G/ is a locally compact, finite dimensional real topological
vector space.
Proof. If we can prove that L.G/ is a topological vector space, then since it
is locally compact by Lemma 7.2, it is finite-dimensional by a theorem of Riesz.
We first prove that scalar multiplication is continuous. Let r 2 R n f0g and let
fr W L.G/ ! L.G/ be defined by fr .X/ D r X. Suppose r 0 2 .r/ and X0 2 .X/.
We must show that r 0 X0 2 .r X/. Since domain.X/ \ . 2; 2/ domain.X0 /,
it is easy to see that domain.r X/ \ . 2; 2/ domain.r 0 X0 /. Furthermore, for
t 2 domain.r X/ \ . 2; 2/ and t 0 2 .t /, one has
.r 0 X0 /.t 0 / D X0 .r 0 t 0 / 2 .X.rt // D ..r X/.t //;
whence r 0 X0 2 .r X/.
We next prove that C is continuous. We first show that C is continuous at
.O; O/. To do this, it suffices to show that for any subbasic open set BC;U of
L.G/ containing O (so that 1 2 U ), there is a subbasic open set BC;W of L.G/
containing O such that BC;W C BC;W BC;U . Let U1 be a compact neighborhood
of 1 such that U1 U . Let Z be an internally open set. Then by Lemma 6.3,
we know that for every a; b 2 Z and for every such that for all i 2 f1; : : : ; g, ai
and b i are defined and ai ; b i 2 Z, we have that .ab/ is defined and .ab/ 2 U1 .
Hence, by transfer, there is an open set W such that for all a; b 2 W and all n with
ai and b i defined and ai ; b i 2 W for all i 2 f1; : : : ; ng, we have .ab/n defined
and .ab/n 2 U1 . We claim that this is the desired W . Suppose X; Y 2 BC;W . Let
t 2 C and suppose is such that t . Then .X C Y/.t / ŒX. 1 /Y. 1 / . Let
a WD X. 1 / and b WD Y. 1 /. By assumption, for all i 2 f1; : : : ; g, ai and b i are
defined and ai ; b i 2 W . Thus, .ab/ 2 U1 , implying that .X C Y/.t / 2 U1 U
and so X C Y 2 BC;U .
In order to finish the proof that C is continuous, it is enough to show that, for
any Y 2 L.G/, the map X 7! X C Y W L.G/ ! L.G/ is continuous at O and the map
X 7! X Y W L.G/ ! L.G/ is continuous at Y. Let W L.G/ be a neighborhood
of Y in L.G/. Fix s 2 domain.Y/ \ . 2; 2/ such that Y.Œ s; s/ U2 . Fix a
compact neighborhood U of 1 in G with U U2 so that NY .s; U / W . Suppose
X 2 .O/ L.G/ . By Lemma 6.2, for all i Œs  we have that .X. 1 /Y. 1 //i
is defined and infinitely close to Y. i /. In particular, this means that for every
X 2 .O/, whenever i Œs , we have ..X. 1 /Y. 1 //i is defined and
i
1
1
i
Y
2Y
U :
X
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
1305
By overspill, this gives a neighborhood V of O in L.G/ such that for all X 2 V
we have that ..X. 1 /Y. 1 //i is defined and
i
1
1
i
X
Y
2Y
U
whenever i Œs. It now follows that for X 2 V ,
t 2 domain.X C Y/ and .X C Y/.t / 2 Y.t /U
for all t 2 Œ s; s, whence X C Y 2 NY .s; U / W . This proves the continuity of
X 7! X C Y at O.
The continuity of X 7! X Y at Y is proved in a similar fashion. Let W L.G/
be a neighborhood of O. Fix s 2 domain.Y/ \ . 2; 2/. Choose a compact neighborhood U of 1 in G with U U2 so that NO .s; U / W . Suppose X 2 .Y/ L.G/ .
By Lemma 6.3, .X. 1 /Y. 1 //i is defined and infinitesimal whenever i Œs . In
particular,
i
1
1
Y
X
2 U
whenever i Œs. By overspill, this gives a neighborhood V of Y in L.G/ such
that for all X 2 V , .X. 1 /Y. 1 //i is defined and
i
1
1
X
Y
2 U
whenever i Œs. It now follows that for X 2 V ,
t 2 domain.X
Y/ and .X
Y/.t / 2 U
for all t 2 Œ s; s, whence X Y 2 NO .s; U / W .
We now need to prove that L.G/ is a vector space. There is only one vector
space axiom which is not trivial to verify, namely that r .X C Y/ D r X C r Y. We
first consider the case that r 2 Z. Let a WD X. 1 / and b WD Y. 1 /. So S.r X Cr Y/ D
.ar b r /G o ./. Meanwhile, S.r .X C Y// D ..ab/r /G o . /. But G. /=G o . / is
abelian, so
S.r .X C Y// D ..ab/r /G o . / D .ar b r /G o . / D S.r X C r Y/:
We have thus proven this axiom for the case r 2 Z.
Once again suppose r 2 Z, r 6D 0. We now know that r . 1r X C 1r Y / D X C Y,
1
i.e., r .X C Y/ D 1r X C 1r Y. So for r 2 Q, we have r .X C Y/ D r X C r Y. We
finally conclude that the axiom holds for all r 2 R from the fact that the operations
of addition and scalar multiplication are continuous.
Remark. Suppose that H and H 0 are locally compact pure local groups. Then
the proof of Theorem 7.5 shows that L.H / is a real topological vector space.
1306
ISAAC GOLDBRING
The NSS assumption was only used to obtain local compactness, and hence finitedimensionality. Moreover, given any local morphism f W H ! H 0 , one can verify
that the map L.f / W L.H / ! L.H 0 / is R-linear and continuous.
C OROLLARY 7.6. G is locally euclidean.
Proof. Since L.G/ is a finite dimensional topological vector space over R, we
have an isomorphism L.G/ Š Rn of real vector spaces for some n, and any such
isomorphism must be a homeomorphism. Hence, by Lemma 7.2, K contains an
open neighborhood U of 1 which is homeomorphic to an open subset of Rn . Shrink
U if necessary so that E.U / W , where W is an open neighborhood of 1 in G
contained in K; such a W exists by Theorem 6.14. Then by Corollary 7.3, E.U /
is an open neighborhood of 1 in G homeomorphic to an open subset of Rn .
8. Local H5 for NSS local groups
The goal of this section is to prove the Local H5 for NSS local groups using
the local version of the Adjoint Representation Theorem.
In this section we assume that G is NSS. Recall that our special neighborhood
has been chosen so that U U6 .
Local adjoint representation theorem. Fix g 2 U6 . Let X 2 L.G/ and X 2 X.
Choose r 2 domain.X / \ R>0 such that X.. r; r// U6 . Define the function
gXg 1 W . r; r/ ! G by the rule .gXg 1 /.t / D gX.t /g 1 . We claim that gXg 1
is a local 1-ps of G. Certainly gXg 1 is continuous. Suppose s; t; s C t 2 . r; r/.
Then since g; g 1 ; X.t /; X.s/ 2 U6 ,
.gXg
1
/.t C s/ D g.X.t /X.s//g
D .gX.t /g
D .gXg
1
1
1
/.gX.s/g
/.t / .gXg
1
1
/
/.s/:
Thus gXg 1 is a local 1-ps of G and Adg .X/ WD g Xg 1 WD ŒgXg 1  2 L.G/ is
easily seen to be independent of the choice of X 2 X and r 2 domain.X / as chosen
above.
L EMMA 8.1. Suppose > N, a 2 G. /, and g 2 U6 . Then
(1) gag
1
2 G./;
(2) Adg .ŒXa / D ŒXgag
1
.
Proof. Let WD ord.a/. Note that D O. /.
C LAIM. If i 2 f1; : : : ; g, then .gag
1 /i
is defined and .gag
1 /i
D gai g
1.
We prove this by internal induction on i . This is certainly true for i D 1 and
we now assume that it holds for a given i and that i C1 . Since we have assumed
that .gag 1 /i is defined, to show .gag 1 /i C1 is defined, it is enough to show, by
Lemma 2.18, that ..gag 1 /k ; .gag 1 /l / 2  for all k; l 2 f1; : : : ; i g such that
1307
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
k C l D i C 1. However, by induction, .gag 1 /k D gak g 1 and .gag 1 /l D
gal g 1 , and since g; ak ; al 2 U6 , we have that .gak g 1 ; gal g 1 / 2  . We thus
know that .gag 1 /i C1 is defined. Now
.gag
1 i C1
/
D .gag
1 i
/ .gag
1
/ D .gai g
1
/.gag
1
/ D gai C1 g
1
since g; ai ; a; g 1 2 U6 .
By the claim, if i Do. /, then .gag 1 /i is defined and .gag 1 /i D gai g 12 since a 2 G./. This proves (1).
We now must prove (2). Fix r 2 R>0 such that Œr  , Xa .. r; r// U6
and gXa .. r; r//g 1 U. Then for t 2 . r; r/,
.Adg .ŒXa /.t // D gXa .t /g
1
D g.st.aŒt  //g
D st.gaŒt  g
1
/
1 Œt 
D st..gag
D Xgag
1
1
/
/
.t /;
which finishes the proof.
The proof of the following lemma is routine.
L EMMA 8.2. For g 2 U6 , Adg W L.G/ ! L.G/ is a vector space automorphism with inverse Adg 1 .
Let Aut.L.G// denote the group of vector space automorphisms of L.G/.
Suppose dimR .L.G// D n. Take an R-linear isomorphism L.G/ Š Rn ; it induces
2
a group isomorphism Aut.L.G// Š GLn .R/ Rn , and we take the topology on
Aut.L.G// that makes this group isomorphism a homeomorphism. (This topology
does not depend on the choice of R-linear isomorphism L.G/ Š Rn .) For T 2
Aut.L.G// , we see that T is nearstandard if T .X/ 2 L.G/ns for all X 2 L.G/.
For T; T 0 2 Aut.L.G//ns , we see that T T 0 if and only if T .X/ T 0 .X/ for all
X 2 L.G/.
T HEOREM 8.3 (Local Adjoint Representation Theorem). We have a morphism
Ad W GjU6 ! Aut.L.G// of local groups given by Ad.g/ WD Adg .
Proof. Suppose g; h; gh 2 U6 and ŒXa  2 L.G/. Then
Adgh .ŒXa / D ŒX.gh/a.gh/ 1  D ŒXg.hah
1 /g
1
 D Adg .Adh .ŒXa //:
Thus Ad.gh/ D Ad.g/ ı Ad.h/. All that remains to prove is that Ad is continuous.
To do this, it suffices to prove that Ad is continuous at 1. Suppose a 2 and
X 2 L.G/.
C LAIM. domain.X/ D domain.Ada .X// \ R.
1308
ISAAC GOLDBRING
In order to verify the claim, suppose . r; r/ domain.X/. We must verify that
if s; t; sCt 2 . r; r/, then .aX.s/a 1 ; aX.t /a 1 / 2  and .aX.s/a 1 /.aX.t /a 1 /
D aX.s C t/a 1 . From the fact that .X.s/; X.t // 2  and a 2 , we have
that .aX.s/a 1 ; aX.t/a 1 / 2  and we have that .aX.s/a 1 /.aX.t /a 1 / D
aX.s C t/a 1 from the usual calculations involving associativity when working
with infinitesimals and nearstandard elements.
Now notice that if t 2 domain.X/, then .Ada .X//.t / D aX.t /a 1 X.t /. So
by Lemma 7.1, Ada .X/ X. By the nonstandard characterization of the topology
on Aut.L.G//, we have Ada idL.G/ , and hence Ad is continuous.
Some facts about local Lie groups. In the global setting, the Adjoint Representation Theorem and some elementary Lie group theory imply that locally compact
NSS groups are Lie groups. In the local group setting, we have to work a little
harder. We need a few ways of obtaining local Lie groups and the results that
follow serve this purpose.
Definition 8.4. A local group G is abelian if there is a neighborhood U of 1
in G such that U U2 and ab D ba for all a; b 2 U .
T HEOREM 8.5. Suppose G is abelian. Then G is locally isomorphic to a Lie
group.
Proof. Suppose we have chosen our special neighborhood U so small that
elements of U commute with each other. Recall that we are still assuming that
U U6 .
C LAIM. Suppose a; b 2 U are such that .ab/n , an , and b n are all defined and
ai ; b i 2 U for all i 2 f1; : : : ; ng. Then .ab/n D an b n .
We prove the claim by induction on n. The case n D 1 is trivial. Now suppose
.ab/nC1 is defined as are anC1 and b nC1 . Also suppose ai and b i are in U for all
i 2 f1; : : : ; n C 1g. Then
.ab/nC1 D .ab/n .ab/ D .an b n / ab D an b n ab D anC1 b nC1 :
This proves the claim.
Choose a symmetric open neighborhood V of O in L.G/ such that V K
and X C Y 2 K for all X; Y 2 V. By the transfer of the claim, if X; Y 2 V, then
1
1
1 1 .X C Y/.1/ D st
X
Y
D st X
Y
D X.1/Y.1/:
After possibly shrinking V, we can choose a symmetric open neighborhood
V of 1 in G such that E.V/ D V . This witnesses that the equivalence class of the
local exponential map is a local isomorphism from L.G/ to G. Since L.G/ is a
Lie group, we are done.
To prove the next lemma, we need the following well-known theorem.
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
1309
T HEOREM 8.6 (von Neumann, [12, p. 82]). If H is a Hausdorff topological group which admits an injective continuous homomorphism into GLn .R/ for
some n, then H is a Lie group.
L EMMA 8.7. Suppose f W G ! GLn .R/ is an injective morphism of local
groups. Then G is a local Lie group.
Proof. Let G 0 D f .G/ GLn .R/. Let H be the subgroup of GLn .R/ generated by G 0 . Let F0 be the filter of neighborhoods of 1 in G and F00 be the image
of F0 under f . Finally, let F be the filter in H generated by F00 . It is routine to
verify that F satisfies the properties needed to make it a neighborhood filter at 1
for a topology on H which makes H a topological group. By the definition of
this topology, the inclusion map H ,! GLn .R/ is continuous. By von-Neumann’s
theorem, H is a Lie group. The result now follows from the fact that G is homeomorphic with f .G/ when f .G/ is given the induced topology from H .
We use the following theorem of Kuranishi [8] to finish our proof of the NSS
case.
T HEOREM 8.8. Let G be a locally compact local group and let H be a normal
sublocal group of G. Consider the local coset space .G=H /W as in Lemma 2.13.
Suppose
(1) H is an abelian local Lie group;
(2) .G=H /W is a local Lie group;
(3) there is a set M W containing 1 and W 0 W , an open neighborhood of
1 in G, such that for every .zH / \ W 2 .W 0 /, there is exactly one a 2 M
such that a 2 .zH / \ W ; moreover, the map .W 0 / ! M that assigns to each
element of .W 0 / the corresponding a in M is continuous.
Then GjW is a local Lie group.
Proof of the NSS case. We can now finish the proof that the Local H5 holds
for NSS local groups. Let G 0 WD GjU6 . Recall the morphism of local groups
Ad W G 0 ! Aut.L.G//. Then H WD ker.Ad/ is a normal sublocal group of G 0 . Since
H is also a locally compact NSS local group, we know that H must have an open
(in H ) neighborhood V of 1 ruled by local 1-parameter subgroups of H , which
are also local 1-parameter subgroups of G. We claim that gh D hg for all g; h 2 V ,
implying that H is abelian. Note that if U0 U is a special neighborhood for H and
if X 2 L.H / is such that I domain.X/ and X.I / U0 , then I domain.g Xg 1 /.
So for g; h 2 V , writing h D X.1/ for some X 2 L.H /, we have that ghg 1 D h.
By Theorem 8.5, we have that a restriction of H is a local Lie group. Let us
abuse notation and denote this restriction, which is an equivalent sublocal group of
G 0 , by H . Note that if we let G 00 WD .G 0 =H /W be some local coset space, then the
adjoint representation induces an injective morphism G 00 ! Aut.L.G//. Thus G 00
1310
ISAAC GOLDBRING
is a local Lie group by Lemma 8.7. In order to finish the proof, we need to show
that condition (3) of Kuranishi’s theorem is satisfied.
Since G 00 is a local Lie group, we can introduce canonical coordinates of the
second kind. More precisely, we can find an open neighborhood U0 of 1 in G 00
and a basis X10 ; : : : ; Xr0 of L.G 00 / such that every element of U0 is of the form
X10 .s1 / Xr0 .sr / for a unique tuple .s1 ; : : : ; sr / 2 Œ ˇ 0 ; ˇ 0 r . Without loss of
generality, we can suppose that the closure of U0 is a special neighborhood of G 00 .
Choose a special neighborhood U of G 0 such that .U/ U0 . Let Z U be an open
neighborhood of 1 ruled by local 1-parameter subgroups of G 0 . Fix s0 2 .0; ˇ 0 /
such that X10 .s0 /; : : : ; Xr0 .s0 / 2 .Z/. Choose xi 2 Z such that .xi / D Xi0 .s0 /.
Let Xi 2 L.G 0 / be such that Xi .s0 / D xi .
Let ˇ < s0 be so that Xi .s/ 2 U2r if jsj ˇ and X1 .s1 / Xr .sr / 2 W if
jsi j ˇ for all i 2 f1; : : : ; rg. A uniqueness of root argument yields that .Xi .s// D
Xi0 .s/ for jsj ˇ. Set
M WD fX1 .s1 / Xr .sr / j jsi j ˇ for all i D 1; : : : ; rg
and let W 0 W be an open neighborhood of 1 in G contained in the image of the
map
.s1 ; : : : ; sr / 7! X1 .s1 / Xr .sr / W Œ ˇ; ˇr ! G:
One can check that the choices of M and W 0 fulfill condition (3) of Kuranishi’s
theorem.
9. Locally Euclidean local groups are NSS
In order to finish the proof of the Local H5, we must now prove that every
locally euclidean local group is NSS. This proceeds in two stages: locally euclidean
local groups are NSCS and locally connected NSCS local groups are NSS.
Locally euclidean local groups are NSCS. We will need the following facts.
T HEOREM 9.1 ([12, p. 105]). Every compact connected nontrivial Hausdorff
topological group has a nontrivial 1-parameter subgroup.
L EMMA 9.2. Let H be a topological group and X W R ! H a 1-parameter
subgroup.
(1) If H1 is a closed subgroup of H , then either X.R/ H1 or there is a neighborhood D of 0 in R such that X.D/ \ H1 D f1g.
(2) If X is nontrivial, then there is a neighborhood D of 0 in R on which X is
injective.
Proof. This is immediate from the well-known fact that a closed subgroup of
the additive group of R different from f0g and R is of the form Zr with r 2 R>0 . The following lemma is the local group version of Theorem 5.1 in [4].
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
1311
L EMMA 9.3. Suppose V is a neighborhood of 1 in G. Then V contains a compact subgroup H of G and a neighborhood W of 1 in G such that every subgroup
of G contained in W is contained in H .
Proof. Let W be an internal neighborhood of 1 in G such that W . Note
that if E1 ; : : : ; E is an internal sequence of internal subgroups of G contained
in W and a1 ; : : : ; a is an internal sequence such that ai 2 Ei for all i 2 f1; : : : ; g,
then a1 a is defined by Lemma 6.1. We let S be the set of all products a1 a ,
where E1 ; : : : ; E is an internal sequence of internal subgroups of G contained in
W and a1 ; : : : ; a is an internal sequence such that ai 2 Ei for all i 2 f1; : : : ; g. By
Lemma 6.1, S is an internal subgroup of G , S , and every internal subgroup
of G contained in W is contained in S . Furthermore, if H is the internal closure
of S in G , then H is an internally compact internal subgroup of G containing
all of the subgroups of W and H V . The desired result follows by transfer. Definition 9.4. Following Kaplansky [7], we call a topological space feebly
finite-dimensional if, for some n, it does not contain a homeomorphic copy of
Œ0; 1n .
Clearly locally euclidean local groups are feebly finite-dimensional.
L EMMA 9.5. If G is feebly finite-dimensional, then G is NSCS.
Proof. Suppose that G is feebly finite-dimensional but, towards a contradiction, that G is not NSCS. We claim that for every compact symmetric neighborhood U of 1 in G contained in U2 and for every n, there is a compact subgroup
of G contained in U which contains a homeomorphic copy of Œ0; 1n . Assume
that this holds for a given n and let U be given. Lemma 9.3 implies that there
is a compact symmetric neighborhood V of 1 in G containing 1 and a compact
subgroup H U that contains every subgroup of G contained in V . Since, by
assumption, V contains a nontrivial connected compact subgroup of G, we have
a nontrivial 1-parameter subgroup X of H . By shrinking V if necessary, we can
suppose that X.R/ ª V . By assumption, we have a compact subgroup G.V / V of
G and a homeomorphism Y W Œ0; 1n ! Y .Œ0; 1n / G.V /. After replacing X by rX
for suitable r 2 .0; 1/, we can assume that X.Œ0; 1/ V , X is injective on Œ0; 1, and
X.Œ0; 1/ \ G.V / D f1g. Since H is a group, we can define Z W Œ0; 1 Œ0; 1n ! H
by Z.s; t/ D X.s/Y.t/, which is clearly continuous. Suppose Z.s; t / D Z.s 0 ; t 0 /
with s s 0 . Then we have X.s s 0 / D Y .t 0 /Y .t / 1 2 X.Œ0; 1/ \ G.V / D f1g.
Since X is injective on Œ0; 1, we have s D s 0 , and since Y is injective, we have
t D t 0 . It follows that Z is injective, and thus a homeomorphism onto its image. Locally connected NSCS local groups are NSS. The next lemma is the local
analog of Lemma 5.3 and Corollary 5.4 in [4]. The proof there has a gap, which
can be filled by changing the hypotheses as in our lemma.
1312
ISAAC GOLDBRING
L EMMA 9.6. Suppose that H is a normal sublocal group of G which is totally
disconnected. Let W G ! G=H be the canonical projection. Then L./ is
surjective.
Proof. Let G 0 WD .G=H /W , where W is as in Lemma 2.13, and let Y 2 L.G 0 /.
We seek X 2 L.G/ so that ı X D Y. If Y is trivial, this is obvious. Thus we
may assume, without loss of generality, that 1 2 domain.Y/ and Y.1/ 6D 1G 0 . Fix
> N and let h WD Y. 1 / 2 .1G 0 /. Take a compact symmetric neighborhood V
of 1G 0 in G 0 such that Y.1/ … V . Take a compact symmetric neighborhood U of
1 in G with U W such that .U / V . Since is an open map, we have that
.1G 0 / ./. Choose a 2 with .a/ D h. Let WD ordU .a/. If , then
.a / 2 V , contradicting the fact that .a / D h D 1G 0 … V . Thus we must have
< . If i D o./, then .st.ai // D st.hi / D 1G 0 , so GU .a/ D fst.ai / j i D o. /g
is a connected subgroup of G contained in H . Since H is totally disconnected, we
must have GU .a/ D f1G g, i.e., a 2 G. /. Since a … G o . /, we have ŒXa  6D O.
Suppose D o./ and let t 2 domain. ı ŒXa /. Then .ŒXa .t // D st.hŒt  / D 1,
whence ŒXa  2 L.H /. Since H is totally disconnected, L.H / is trivial and hence
ŒXa  D O, a contradiction. Thus we have D .r C "/ for some r 2 R>0 and
infinitesimal " 2 R . Thus ı ŒXa  D r Y and X WD 1r ŒXa  is the desired lift of Y.
We need one fact from the global setting that we include here for completeness.
It is taken from [4].
L EMMA 9.7. Suppose G is a pure topological group such that there are no
nontrivial 1-parameter subgroups X W R ! G. Then G has a neighborhood base
at 1 of open subgroups of G. In particular, G is totally disconnected.
Since locally euclidean local groups are locally connected, the next theorem,
in combination with Lemma 9.5, finishes the proof of the Local H5.
T HEOREM 9.8. If G is locally connected and NSCS, then G is NSS.
Proof. Let V be a compact symmetric neighborhood of 1 in G such that V
contains no nontrivial connected subgroups. Choose an open neighborhood W of 1
in G with W V and a compact subgroup H1 of G with H1 V with the property
that every subgroup of G contained in W is contained in H1 ; these are guaranteed
to exist by Lemma 9.3. Without loss of generality, we can assume that W U6 .
We now observe that if a 2 is degenerate, then a 2 H1 , for the internal subgroup
internally generated by a is contained in W . Since G is pure, if a 2 is
nondegenerate, then it is pure; for such an a, if a 2 H1 , then a 2 H1 V for all , contradicting the fact that some power of a lives outside of V . Hence,
nondegenerate infinitesimals live outside of H1 .
Since H1 V , H1 admits no nontrivial 1-parameter subgroups, and hence H1
must be totally disconnected by Lemma 9.7. Choose an open (in H1 ) subgroup H
of H1 contained in H1 \ W . Write H D H1 \ W1 with W1 an open neighborhood
HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
1313
of 1 in G and W1 W . Since H is an open subgroup of H1 , it is also closed in
H1 , and thus H is a compact subset of G. This implies that the set
U WD fa 2 W1 j aHa
1
W1 g
is open. Fix a 2 U . Since aHa 1 is a subgroup of G contained in W1 W , we
have aHa 1 H1 . Thus aHa 1 H1 \ W1 D H implying that H , considered
as a sublocal group of G, is normal. Replacing U by U \ U 1 , we can take U as
the associated normalizing neighborhood for H .
Note that H U6 and H 6 U . Find symmetric open neighborhoods W2
and W3 of 1 in G so that H W3 W3 W2 and so that W2 U6 and W26 U ;
this is possible by the regularity of G as a topological space and the fact that H is
compact. Let G 0 WD .G=H /W2 . Suppose a 2 . If a is degenerate, then we have that
a 2 H1 \ W1 D H , whence .a/ D 1G 0 . Now suppose that a is nondegenerate.
Let WD ordW3 .a/. Then a 2 .W3 / , but a C1 2 W2 n .W3 / . Choose an open
neighborhood U 0 of 1 in G so that U 0 H W3 . Suppose .a C1 / D .x/ for some
x 2 .U 0 / . Then a C1 D xh for some h 2 H , whence a C1 2 W3 , a contradiction.
Hence .a/ C1 … ..U 0 / /. Meanwhile, for i D o. /, we have .a/i 2 .G 0 /.
Thus .a/ is pure in G 0 and we have shown that G=H has no degenerate infinitesimals other than 1G 0 ; in other words, we have shown that G 0 is NSS.
Since G and G 0 are pure, L./ W L.G/ ! L.G 0 / is a continuous R-linear
map. Since H is totally disconnected, L./ is surjective by Lemma 9.6. Note that
if X 2 ker.L.//, then X can be considered a local 1-ps of H . Since H is totally
disconnected, L.H / is trivial, and hence X D O. Since L.G 0 / is finite-dimensional,
we can conclude that the map L./ is an isomorphism of real topological vector
spaces.
Now take a special neighborhood U0 of G 0 and let E 0 W K0 ! K 0 denote the
local exponential map for G 0 . Take a connected neighborhood U of 1 in G such that
U W2 and .U/ K 0 . Let x 2 U. Since E 0 is a bijection, there is a unique Y 2 K0
such that .x/ D E.Y/. Since L./ is a bijection, there is a unique X 2 L.G/
such that ı X D Y. Thus we can write x D X.1/x.H / where x.H / 2 H . One
can easily verify that the map which assigns to each x 2 U the above Y 2 L.G 0 / is
continuous. From this and the fact that L./ is a homeomorphism, we see that the
map x 7! x.H / W U ! H is continuous. Since U is connected, H is totally disconnected, and 1.H / D 1, it follows that x.H / D 1 for all x 2 U. Now by the injectivity
of L./ and E 0 , we have that jU is injective, implying that G cannot have any
subgroups other than f1g contained in U. We thus conclude that G is NSS.
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(Received June 22, 2007)
(Revised July 16, 2007)
E-mail address: isaac@math.ucla.edu
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table of contents
Gonzalo Contreras. Geodesic flows with positive topological entropy,
twist maps and hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761–808
Eckart Viehweg. Compactifications of smooth families and of moduli
spaces of polarized manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809–910
Chang-Shou Lin and Chin-Lung Wang. Elliptic functions, Green
functions and the mean field equations on tori . . . . . . . . . . . . . . . . . . . . . . . . . . . 911–954
Yichao Tian. Canonical subgroups of Barsotti-Tate groups . . . . . . . . . . . . . . . . 955–988
Akshay Venkatesh. Sparse equidistribution problems, period bounds
and subconvexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989–1094
Thomas Geisser. Duality via cycle complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 1095–1126
Viktor L. Ginzburg. The Conley conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127–1180
Christopher Voll. Functional equations for zeta functions of groups
and rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1181–1218
Monika Ludwig and Matthias Reitzner. A classification of SL.n/
invariant valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1219–1267
Isaac Goldbring. Hilbert’s fifth problem for local groups. . . . . . . . . . . . . . . 1269–1314
Robert M. Guralnick and Michael E. Zieve. Polynomials with
PSL.2/ monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315–1359
Robert M. Guralnick, Joel Rosenberg and Michael E. Zieve. A
new family of exceptional polynomials in characteristic two . . . . . . . . . . . . 1361–1390
Raman Parimala and V. Suresh. The u-invariant of the function fields
of p-adic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1391–1405
Avraham Aizenbud, Dmitry Gourevitch, Stephen Rallis and
Gérard Schiffmann. Multiplicity one theorems . . . . . . . . . . . . . . . . . . . . 1407–1434
Stanislav Smirnov. Conformal invariance in random cluster models.
I. Holmorphic fermions in the Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435–1467
Kannan Soundararajan. Weak subconvexity for central values of
L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1469–1498
Roman Holowinsky. Sieving for mass equidistribution . . . . . . . . . . . . . . . . . 1499–1516
Roman Holowinsky and Kannan Soundararajan. Mass
equidistribution for Hecke eigenforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517–1528
Kannan Soundararajan. Quantum unique ergodicity for
SL2 .Z/nH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1529–1538